Last Update : 23 February 2021
Geometrikon is a gallery of Topics in Geometry and geometric subjects from other fields. It started as a list of examples from Geometry, Euclidean, Affine, and Projective,
and evolved for some time to contain proofs and discussions on various topics. The subjects listed below are continuously reviewed and updated
with the intention to make the exposition as complete as possible and give enough references to related subjects and bibliography.
Originally the gallery included several hundreds of topics, some rudimentary and intentional for future discussions, some informal, some included for
aesthetic reasons, some in the form of detailed expositions with proofs.
Presently the old html-files of the gallery, representing the topics, are gradually replaced with pdf-files and updated contents. The aim for this is
a stronger coherence in the presentation of material and a better quality of figures.
To achieve this, the presentation of the topics is organized as a collection of possibly independent pdf-files.
The topics below are listed alphabetically using some relevant for their content word of the title.
To search quickly Geometrikon for
an item of your interest, display the "search box" by pressing the key-combination CTRL+F, write there the search word and press the OK button or the ENTER-key.
The "hyper-references" in the new versions with pdf-files, which refer to other pdf-fils are "relative" and work properly in "firefox" if you read them on-site.
They work correctly off-site, if you download the respective files and put them all in the same folder.
If you like to be informed on updates and new links to pdf-files of this gallery, send me an email to add you in the list of updated-urls-recipients.
- Abridged notation [New pdf-version] 01/09/2009, 28/11/2018 , 10/12/21
Contents:
1 Introduction ,
2 Abridged notation, cartesian coordinates, homogenization ,
3 Pencils of conicss ,
4 Interpretation of the members of a pencil ,
5 Bitangent conics and pencils ,
6 Abridged notation, line formularium ,
7 Abridged notation, conics formularium ,
8 Cross ratio on the conic ,
9 The Chasles-Steiner definition of a conic ,
10 Some applications, Pascal’s theorem ,
11 Maclaurin’s conic generation method ,
12 Double tangency of two conics ,
13 Polygons inscribed and circumscribed about conics .
Related topics :
Cross Ratio,
Homographic relation,
Projective line.
Projective plane.
The quadratic equation in the plane.
- Affine transformations of the plane (Affinities) [New pdf-version] 01/09/2009, 14/05/2019
Contents:
1 Definition and group property,
2 Determination of an affinity,
3 Additional fundamental properties of affinities,
4 Special kinds of affinities,
5 Axial affinities or homologies, shears, strains,
6 Affinities leaving invariant a triangle,
7 The period of an affinity,
8 Translations,
9 Dilatations,
10 Shears,
11 Strains,
12 Invariant pencils,
13 Equiaffinities as products of reflections,
14 Conjugacy for affinities with fixed points,
15 Elliptic affinities,
16 Orbits of points under equiaffinities,
17 Classification of equiaffinities,
18 Analytic description of orbits of equiaffinities,
19 Remarks and exercises on orbits of equiaffinities,
20 Affine equivalence of conics,
21 Affinities preserving a non-degenerate conic.
- Agnesi-curve
Illustration of the curve "Witch of Agnesi" and description of its generation.
- Algebraic_Locus
A locus generated by some algebraic operations on (complex) points.
- All Conics Circumscribed created : 27/10/2007, updated : 27/08/2009 euc
Determination of the family of all conics circumscribing a triangle and tangent to a line.
- All Conics Circumscribed (2)
As before but now the line separating the three vertices of the triangle.
- Conics circumscribing a triangle
A study of the conics circumscribed about a triangle.
- All Parabolas Circumscribed created : 22/06/2006, updated : 31/01/2008, 27/08/09 euc
Determination of the family of all parabolas circumscribing a triangle. Some remarks on the two parabolas through four points.
- Lines through a fixed point
A property of lines defined through ratios on the sides of an angle.
- Lines defined through ratios
Defining a line through certain length-ratios on the sides of an angle.
- AnalyticVersusSynthetic 15/12/2009, 5/2/10 euc
The merits of analytic and synthetic methods of proof in geometry. Study on the ground of an example (locus).
- Angle problem 5/01/2010 euc
A gem in determining the angles of a simple figure constructed in an isosceles triangle.
- Angles on sides of a triangle
Constructing equal angles on the sides of a triangle. A geometric locus and remarks on Brocard's points.
- Angle Multiples
A tutorial on the various ways to measure angles and their applications.
- Anticomplementary and Circumparabola created : 12/02/2008, updated : 27/08/09 euc
A very simple relation of the anticomplementary triangle with a circumparabola of the triangle.
- Antiparallel Hyperbola created : 31/12/2007, updated : 14/02/2008 euc
A hyperbola connected to the antiparallels to a side of a triangle.
- Apollonian circles of a segment, Apollonian pencil [New pdf-version] 10/05/2005, 27/05/2019, 25/11/2021
Contents:
1 Apollonian circles of a segment,
2 Inversion properties of Apollonian circles,
3 The Apollonian pencil of a segment,
4 Related exercises.
Related topics :
Circle pencils,
Inversion,
Isodynamic points.
Pedal triangles of a triangle.
- Apollonian circles of a triangle and Isodynamic points
[New pdf-version] 17/01/2010, 27/5/2019, 16/09/2021
Contents:
1 Apollonian circles of a triangle,
2 Isodynamic points, Brocard and Lemoine axes,
3 Given Apollonian circles,
4 Given isodynamic points,
5 Trilinear coordinates of the isodynamic points,
6 Same isodynamic points and same circumcircle,
7 A matter of uniqueness.
Related topics :
Apollonian circles,
Circle pencils,
Inversion,
- Apollonius Problem [New pdf-version] 10/05/2005, 16/11/2018, 25/11/2021
Contents:
1 Apollonius problem,
2 Two points and a line, category (0,1,2) ,
3 Two points and a circle, category (1,0,2) ,
4 Two lines a point, category (0,2,1) ,
5 One circle one line one point, category (1,1,1) ,
6 Two circles one point, category (2,0,1) ,
7 Three circles, category (3,0,0) ,
8 Soddy circles ,
9 Pictures of various cases.
Related topics :
Circle pencils,
Inversion,
Power,
Tritangent circles,
- Arbelos
Shoemaker knife and some properties related.
- Arc-polygon inscribed in a triangle
Study of an arc-hexagon inscribed in a triangle, related to the Gergonne point of the triangle.
- Arc-hexagon inscribed in a polygon
Study of an arc-hexagon inscribed in another circumscriptible hexagon.
- Arc-hexagon inscribed in a polygon (2)
Continuation of the previous study, with properties related to the diagonals of the polygons.
- Arc-pentagon inscribed in a polygon
Study of an arc-pentagon inscribed in another circumscriptible pentagon.
- Arc-polygon (path) in a quadrangle
A picture of an arc-polygon inscribed in a general quadrangle.
- Oscillation of a point on an arc
How to construct with EucliDraw an oscillating point on a circular arc > 180 degrees.
- Archimedes' Circles
Two equal in radii circles tangent to the circles of an arbelos.
- Area in Barycentric coordinates created : 31/12/2007, updated : 15/12/2009 euc
The area of a triangle expressed in the barycentric coordinates of its vertices.
- Area through determinant created : 31/12/2007, updated : 15/12/2009 euc
The expression of the area of a triangle/polygon in terms of the determinant of the coordinates of its vertices.
- Area Quotient
Comparing the areas of a parallelogram and of the quadrangle, which results by reflecting a point on the sides of the parallelogram.
- AreasRatio
Comparing the areas of a triangle and an inscribed to it other triangle.
- Artobolevsky's ellipsograph
A simple link-gear mechanism to draw ellipses and their tangents.
- Artzt parabolas of first kind created : 02/02/2007, updated : 10/10/2008 euc
Three parabolas associated to a triangle and their relation to the second Brocard triangle.
- The three Artzt parabolas of a triangle created : 31/01/2008, updated : 10/10/2008 euc
Three parabolas associated to a triangle and their relation to the second Brocard triangle.
- Artzt parabola generation III
The simplest case of an Artzt parabola and its transfer to the general case through affinities.
- Artzt parabola generation
Generating an Artzt parabola as envelope of diagonals of parallelograms (projective coordinates proof).
- Artzt parabola generation II
Generating an Artzt parabola as envelope of diagonals of parallelograms (synthetic proof).
- Artzt parabolas of second kind
Three more parabolas associated to a triangle and their relation to the second Brocard triangle.
- Artzt parabola of an isosceles triangle created : 10/10/2008 euc
A parabola associated naturally to an isosceles triangle and a way to generate it through line-intersections.
- Generating conics by using the Artzt parabola created : 10/10/2008 euc
Use the Artzt parabola of an isosceles triangle to generate various conics through line intersections.
- Parabola to circle projectivities created : 10/10/2008 euc
Study of a connection of the parabola with the circle-pencils on some chords of it.
- Artzt-parabolas connection to outer Steiner ellipse created : 10/10/2008 euc
A natural connection of the three Artzt parabolas of a triangle to the outer Steiner ellipse of it.
- Three conics naturally associated to a circumconic created : 10/10/2008 euc
A natural connection of the three conics and a triangle conic with a given perspector.
- Intersection point of Artzt parabola with inner Steiner conic created : 1/2/2008 euc
The intersection point of an Artzt parabola of first kind with the inner Steiner conic.
- Asymptotic triangle
Some properties of the asymptotic triangles of a hyperbola.
- Asymptotic triangle inverse created : 22/06/2006, updated : 05/05/2008 euc
The inverse construction of a hyperbola from one of its asymptotic triangles.
- Autopolar or self-polar triangle
A triangle possessing a circle with respect to which every side is the polar of the opposite vertex.
- Autopolar or self-polar triangle - II created : 05/05/2008 euc
A natural way to produce autopolar or self-polar triangles with respect to a conic. Fundamental properties of such systems.
- Auxiliary Circle
The circle with diameter the major axis of an ellipse. Its basic properties.
- Barycentric coordinates [New pdf-version]
01/09/2009, 01/01/2019, 09/06/2021, 10/09/24
Contents:
1 Preliminaries and definition ,
2 Traces, ratios, harmonic conjugation ,
3 Lines in barycentric coordinates ,
4 Ceva’s and Menelaus’ theorems in barycentrics ,
5 Trilinear polar in barycentrics ,
6 Relation to cartesian coordinates, inner product ,
7 The circumcircle of ABC in barycentrics ,
8 Displacement vectors, inner product, distance ,
9 Orthogonality of lines , orthocentroidal circle
10 Distance of a point from a line , distance of two parallels
11 Meaning of line coefficients ,
12 Power of a point, general circle , Euler circle
13 Centroid, Incenter, Circumcenter, Symmedian point ,
14 Euler line, Orthocenter, center of Euler’s circle ,
15 Triangle Area in Barycentrics ,
16 Circumcevian triangle of a point ,
17 Circle through three points , Brocard circle
18 The associated affine transformation ,
19 Relations between barycentrics and cartesian coordinates ,
20 Affine transformations represented in barycentrics ,
21 Polar and and de Lonchamps circles ,
22 Remarks on working with barycentrics ,
23 The Steiner point of a triangle ,
Related topics :
Conway triangle symbols,
Cross Ratio,
Harmonic,
Projective Line.
- Basic Quadrangle Identity
An identity valid for every quadrangle.
- Basic Quadrangle Identity (2)
A second identity valid for every quadrangle.
- Bellows
Slider-Crank triple bellows mechanism, animated through motor-objects of EucliDraw.
- Bellows2
Slider-Crank quadripartite bellows mechanism , animated through motor-objects of EucliDraw.
- Bezier
Illustration of a user-defined-tool creating a Bezier arc with four control-points.
- General Bezier
Illustration of a user-defined-tool creating a General Bezier arc with N control-points.
- Rational Bezier
Illustration of a user-defined-tool creating a Rational Bezier arc with N control-points and N weights.
- Bicentric
Basic properties of quadrangles that have both inscribed and circumscribed circles.
- Bicentric rectangular hyperbolas
Inverting the circles of a pencil involved with bicentric quadrangles.
- Bicentric Loci
Geometric Loci connected with bicentric quadrilaterals.
- Bicorn curve 5/2/2010 euc
A curve similar to Napoleon's hat appearing when dynamically modifying the classical Pythagora's figure.
- Billiard ball trajectories
Illustration of a user-defined-tool creating Billiard ball trajectories inside an arbitrary polygon.
- Billiard ball closed trajectories in triangles
Example of construction of a closed billiard ball trajectory inside a triangle.
- Billiard ball trajectories in a triangle
Illustration of a user-defined-tool creating Billiard ball trajectories inside an arbitrary triangle.
- Bisecting angle of tangents created : 22/06/2006, updated : 15/12/2009 euc
Bisecting the angle of two tangents to a conic from a point. A theorem of Steiner.
- Bisector-I created : 22/06/2006, updated : 02/02/2011
The basic figure of bisectors of a triangle and some of its properties.
- Bisector-II created : 22/06/2006, updated : 10/10/2008, 02/02/2011 euc
The basic figure of bisectors of a triangle and some further properties.
- Bisector-III
Some consequences of the properties examined in the two previous files.
- Bisector Cross
A cyclic quadrangle divided in triangles through its diagonals. A nice cross from the incircles of these triangles.
- Bisector Parabola
A parabola associated naturally to a bisector of a triangle.
- Bisector Ratio property
A property of rectangular hyperbolas characterizing them as locus of vertices of triangles with bisectors of fixed ratio.
- Bisector Rectangle
A rectangle created by the bisectors of two triangles inscribed in the same circle.
- Bitangent Conics created : 31/12/2007, updated : 28/08/2009 euc
Study of a family of conics and the corresponding homographies preserving its members.
- Bitangent rectangular member
The rectangular hyperbola member of a bitangent family of conics.
- Bitangent rectangular member II
The rectangular hyperbola member of a bitangent family of conics, additional remarks.
- Bolzano Weierstrass application
Applying the theorem of Bolzano and Weierstrass to prove Kroneker's theorem.
- Box Fractal
A fractal resulting by repeatedly dividing a square in 9 equal squares.
- Braikenridge Theorem created : 15/12/2009 euc
Braikenridge's theorem: converse of Maclaurin's theorem.
- Brianchon's theorem
Brianchon's theorem on hexagons circumscriptibles in circles.
- Brianchon's theorem for pentagons and quadrangles created : 22/06/2006, updated : 05/05/2008 euc
Brianchon's theorem specialized for pentagons and quadrangles circumscriptible to circles.
- Brianchon's theorem for conics created : 22/06/2006, updated : 05/05/2008 euc
Brianchon's theorem on hexagons circumscriptibles in conic sections.
- Bricard exact straight-line mechanism
Illustration of a linkage-mechanism, transforming circular motion to straight-line motion exactly.
- Brocard points, Brocard angle
Definition and first properties of the two Brocard points and the Brocard angle of a triangle.
- Brocard points of pivoting triangles
Some properties of the triangles created from the basic configuration, defining the Brocard points.
- Brocard Ellipse
A quick introduction to the Brocard ellipse of a triangle.
- Brocard's second triangle
Definition and first properties of Brocard's second triangle of a triangle ABC.
- Butterfly theorem [New pdf-version] 29/04/2006, 27/11/2018
Related topics :
Cross Ratio,
Cyclic Projective,
Harmonic,
Projective Line,
- pencil Quadrangle
Some quadrangles related to circle pencils and their properties.
- Carnot's problem
Finding a polygon by prescribing the middles of its sides.
- Carnot's theorem created : 15/12/2009 euc
Carnot's theorem on the intersections of triangle-sides with a conic.
- Castillon's problem
Finding a polygon inscribed in a given conic and having its sides passing through given points.
- Castillon's problem in circles
A special case of the previous. Finding a polygon inscribed in a given circle and having its sides passing through given points.
- Catenary
A remarkable curve created by free hanging catenaries.
- Centroid and related remarks
Some remarks on the centroid of a triangle and questions related to triangle centers.
- Centroid (Barycenter or center of gravity)
A property of centroids using vector operations.
- Centroid's basic property
A property of centroids that relies on vector addition.
- Ceva's theorem [New pdf-version] 24/09/2010, 02/12/2018, 15/12/2021
Contents:
1 Ceva’s theorem ,
2 Cevians,
3 Relation of Ceva’s and Menelaus’ theorems ,
4 A limit case ,
5 A second version of Ceva’s theorem ,
6 Vectorial form of Ceva’s theorem ,
7 Projective version of Ceva’s theorem ,
8 Projective version using an arbitrary line,
9 Triangle’s ratio coordinates,
10 Dividing the sides of a triangle.
Related topics :
Apollonian circles,
Barycentric coordinates,
Cross Ratio,
Desargues' theorem,
Isodynamic points of the triangle,
Menelaus' theorem,
Nagel center of the triangle,
Projective line,
Symmedian center of the triangle,
Tritangent circles of the triangle.
- Cevian Bisectors
Definition and first properties of an ellipse defined naturally through the cevians of a point.
- Cevians and parallels
When the parallels to the sides define a triangle perspective to the cevian triangle of a point.
- Chasles - Steiner conics created : 22/06/2006, updated : 05/05/2008, 10/10/2008, 02/02/2011 euc
The generation of conics through homographic correspondence of two pencils of lines.
- Chasles - Steiner conics II created : 10/10/2008 euc
The generation of conics through homographic correspondence of two pencils of lines, additional remarks.
- Chasles - Steiner conics III created : 10/10/2008 euc
The inverse procedure to the Chasles-Steiner generation, producing a homographic relation out of a degenerate conic.
- Chasles - Steiner line enveloping conics
The dual to the previous generation of conics as envelopes of lines connecting points on two lines related by a homography.
- Chasles - Steiner example created : 05/05/2008 euc
Example of generation of a conic by the Chasles-Steiner procedure.
- Chebyshev's linkage
Illustration of a linkage-mechanism, due to Chebyshev, giving four-bar approximate straight-line mechanism.
- Circle Pencils [New pdf-version] 02/08/2006, 15/02/2019, 12/12/21
Contents:
1 The two main types of pencils ,
2 The tangential type of pencils ,
3 Pencils described by equations,
4 Common properties to all types of pencils ,
5 Orthogonal circles ,
6 Orthogonal pencils ,
7 Three exeptional types of pencils ,
8 Reduction to exceptional cases ,
9 Two prominent pencils ,
10 Polars and poles w.r. to members of a pencil ,
11 Quadratic transformation defined by a pencil ,
13 Hyperbola from a pencil of circles ,
14 Rectangular hyperbolas related to pencils of circles ,
15 Some exercises related to pencils of circles.
Related topics :
Apolloninan circles ,
Homographic relations,
Inversion,
- Circles Similarity Centers created : 22/06/2006, updated : 27/08/09 euc
Location of the similarity centers of two circles and a related problem.
- Circular Fractal
Circular Fractal, driving EucliDraw to its limits.
- Circle Tangents Properties created : 02/02/2011
Study of certain properties of the tangents to a circle from a point.
- Circle Centers created : 02/02/2011
Study of the locus of centers of circles simultaneously tangent to two circles.
- Circular_Cubics
Circular cubics, created by the intersection-points of two "lines" of circles (circle-pencils).
- Circumcenter
Circumcenter of a triangle.
- Circumcenter Locus
Circumcenter Locus of a triangle inscribed in an ellipse. Parametric equation of the locus.
- Circumcircle-conjugate (Brocard ellipse)
The Brocard ellipse of a triangle as the conjugate of the circumcircle.
- Circumcircle In Trilinear Coordinates
The equation of the circumcircle of a triangle in trilinear and barycentric coordinates.
- Circumcircle Properties created : 12/02/2008 euc
Properties of projections of a circumcircle point on the sides of triangles and quadrangles.
- Circumconics Generation created : 28/08/2009 euc
Generation of circumconics by secants through the harmonic associates of the pivot.
- Circumconics Tangent Generation created : 12/02/2008 euc
Generation of circumconics tangent to a given line through a point moving on the given line and associated line intersections.
The structure of the set of circumconics which are also tangent to a given line.
- Circumconics tangents created : 22/06/2006, updated : 05/05/2008 euc
Generation of a circumconic of a triangle as envelope of trilinear polars.
- Circumconics Tangent to a Line created : 12/02/2008 euc
The structure of the set of circumconics which are also tangent to a given line.
- Circumconics of a Trapezium created : 5/2/2010 euc
A picture of the family of conics passing through the vertices of a trapezium.
- Circummedial triangle
The triangle intercepted by the medians on the circumcircle of a triangle.
- Circumparabola Generation created : 12/02/2008 euc
A simple way to generate a circumparabola through line intersections.
- Circumscriptible quadrangle created : 22/06/2006, updated : 05/05/2008 euc
Proof of the necessary and suficient condition for a quadrangle to be circumscriptible.
- Circumscriptible Construction
How to construct all circumscriptible quadrangles with given sides.
- Circumscriptible Quadrilateral created : 22/06/2006, updated : 05/05/2008 euc
Quadrilateral circumsctiptible on circle and a basic property of the intersection of its diagonals.
- Circumscriptible Quadrilateral II created : 22/06/2006, updated : 05/05/2008 euc
Quadrilateral circumsctiptible on circle. Basic intersection-properties for its diagonals.
- Cleaver : bisecting the perimeter of a triangle
Bisecting the perimeter of a triangle through a line. Variations on a theorem of Archimedes.
- Clifford-Cayley Theorem created : 22/06/2006, updated : 12/12/2009 euc
Clifford-Cayley Theorem on three linked similar triangles.
- Closed polygonal lines
Closed polygonal lines inscribed in circles and recurring to their start point.
- Closed polygonal lines in conics
Closed polygonal lines inscribed in conics and recurring to their start point.
- Complete Quadrangle
Complete quadrangle, its definition and basic properties.
- Common polars
Points on the major axis of the ellipse have the same polar w.r to it and the auxiliary circle.
- Common polars consequences
The polars of a point w.r. to the ellipse and its auxiliary circle intersect on its major axis.
- Conchoid of Circle
Conchoid of Circle, how it is generated mechanically.
- Conic characterization 15/12/2009 euc
Parameterization of conics through quadratic rational functions. Determination of geometric characteristics.
- Conic construction 15/12/2009 euc
From two tangents, its contact point with one of the tangents and one of its foci.
- Conic equation 5/2/2010 euc
A study of the conic equation in the setting of vectors and matrices.
- Conic from three points and two tangents
Construction of a conic passing through three points and having two given tangents.
- Conic from three points and two tangents-II
Construction of a conic passing through three points and having two given tangents, second general case.
- Conic Homographies 22/06/2006, 28/08/2009 euc
On homographies of preserving a conic.
- Conic instrument 5/2/2010
A file allowing the dynamic change of the six coefficients determining a conic in a cartesian coordinate system.
- Conics and similarities
Conics resulting by a similarity between two circles.
- Conics and similarities II
Conics resulting by a similarity between two circles. A case of non-tangency to the two defining circles.
- Conics and similarities III
Conics resulting by a similarity between two circles. A case of non-tangency to both defining circles.
- Conics degenerate created : 15/12/2009 euc
Variations on a set of two lines related to degenerate conics: degenerate or reducible conics.
- Families of conics
A first view on families of conics.
- Conics from circles
Conics as geometric loci of points equidistant from two circles.
- Conic intersections with two variable lines 5/2/2010 euc
Description of a line defined by intersecting a conic through two other variable lines. Application to the Fregier point.
- Conic passing through origin 5/2/2010 euc
A picture of the conic passing through the origin. A related family of conics through the vertices of a trapezium.
- Conics in homogeneous coordinates 5/2/2010 euc
An introduction to the study of projective conics using homogeneous projective coordinates.
- Conics as line-envelopes
Generalization of a well known theorem, representing a conic as line-envelope.
- Conics Maclaurin
A basic method of generation of conics through intersections of varying line-pairs, illustrated in the trivial case of an equilateral triangle.
- Conics Maclaurin II created : 22/06/2006, updated : 28/08/2009 euc
A basic method of generation of conics through intersections of varying line-pairs, reducing the general case to the trivial one.
- Conics Maclaurin III
The locus of perspectors of Maclaurin conics passing through three points.
- Conics Tangent and intersecting created : 05/05/2008 euc
Some properties of conics tangent at a point and intersecting at two other points.
- Conics defined through rational Bezier curves
Illustration of a user tool producing arcs of arbitrary conics using rational Bezier curves of degree 2.
- Conics through four points geometric created : 05/05/2008 euc
Conics passing through four points considered as triangle conics.
- Conics and quadratic parameterizations created : 15/12/2009 euc
Generation of conics through quadratic rational functions along two arbitrary lines.
- Conjugate diameters homography 5/2/2010 euc
A homography defined on every ellipse through the conjugation of its diameters.
- Conjugate diameter involution
Conjugate diameter involution on the points of a conic.
- Conjugacy with respect to conics created : 28/08/2009 euc
Conjugacy of points/lines with respect to a conic and some applications.
- A problem on inscribed angles
A problem on inscribed angles on two intersecting circles.
- Domain of convexity for quadrangles (entire)
Solution to the problem: Which points produce with their reflections on the sides of a quadrangle a convex quadrangle.
- Conway triangle symbols 02/08/2006, 06/11/2018, 23/02/2021
Contents:
1 Definition, first properties ,
2 Identities resulting directly from definition ,
3 Connection with the Brocard angle ,
4 Playing with the formulas, GH, GI ,
5 Euler’s theorem, Gerretsen’s inequalities ,
6 Connection with the fundamental invariants ,
7 Orthic axis = Radical axis of circumcircle and Euler circle ,
8 GHI triangle, Feuerbach point ,
9 Third degree equation for the symbols.
Related topics :
Barycentric coordinates ,
Cross Ratio ,
Fundamental invariants of the triangle ,
Nagel point of the triangle.
Projective Line.
Tritangent circles.
- Coordinates Basics created : 22/06/2006, updated : 31/01/2008, 28/08/2009 euc
Basic relations in oblique and normal coordinate systems of the euclidean plane.
- Coordinates transform created : 12/12/2007, updated 05/05/2008, 28/08/2009 euc
Study of a particular coordinate transformation between two ways to project a point on two lines.
- Coordinates transform II created : 12/12/2007, updated 05/05/2008 euc
Study of a particular coordinate transformation between two ways to project a point on two lines. A variant on the previous one.
- Ellipse of cosine theorem
An ellipse related to the cosine theorem.
- Cosymmedian triangle
The triangle with vertices the traces of the cosymmedians on the circumcircle of a triangle.
- Cross Ratio [New pdf-version] 22/06/2006, 31/01/2008, 24/06/2019, 09/02/2021
Contents:
1 Cross Ratio ,
2 Properties of cross-ratio of four numbers ,
3 Cross ratio expressed through angles ,
4 Harmonic pencils of lines ,
5 Coincidences and cross ratios of pencils ,
6 Cross ratio on a circle ,
7 Cross Ratio on a conic ,
8 Cross Ratio Formularium ,
9 Complex Cross Ratio ,
10 Circle parametrization through the cross ratio ,
11 Further properties of the cross ratio.
Related topics :
Ceva's theorem ,
Homographic relation ,
Menelaus' theorem ,
Projective Line.
- Cubic through four points
Construction of a parametric cubic passing through four given points.
- Cubic from two conics created : 05/05/2008 euc
Construction of a nodal cubic starting from two tangent and simultaneously intersecting conics.
- Cubic, its graphic solution created : 02/02/2011
Find the roots of a cubic equation using a parabola and a rectangular hyperbola.
- Cubic Reduced created : 05/02/2008, updated 02/02/2011
The geometric properties of the cubic in reduced form: y = x^{3} + px - q and an application to the so-called fundamental invariants of a triangle.
- Cubic Symmetry
The symmetry of the graph of the general cubic function y = ax^{3}+bx^{2}+cx +d about its inflexion.
- Cubic With Complex Roots
Some interesting geometric relations of the cubic y = ax^{3}+bx^{2}+cx +d when it has two complex conjugate roots.
- Cut under a given angle
With a line cut another line or circle under a given angle.
- Cyclic quadrilateral
Some elementary properties of the cyclic quadrilateral.
- Maximizing property for cyclic quadrangles
Cyclic quadrangles have maximum area among all quadrangles with the same sides.
- Cyclic quadrilateral from projective viewpoint
How the general projective properties of a quadrangle specialize in the cyclic case.
- Cycloid
Well known curve, produced by a point attached to a wheel moving on a line.
- Delone Linkage
Delone slider-crank mechanism for tracing ellipses.
- De Longchamps ellipse created : 10/10/2008 euc
De Longchamps ellipse of a triangle.
- Deltoid created : 10/10/2008 euc
Deltoid naturally associated to a triangle.
- Deltoid basic overloaded picture created : 10/10/2008 euc
An overloaded picture of the Deltoid associated to a triangle.
- Derivative triangle of another triangle created : 10/10/2008 euc
An equilateral triangle naturally associated to a triangle and its deltoid.
- Desargues' theorem and perspectivities [New pdf-version] 12/12/2007, 28/08/09, 10/01/12, 29/06/19
Contents:
1 Desargues’ theorem ,
2 Perspective triangles ,
3 Desargues’ theorem, special cases ,
4 Sides passing through collinear points ,
5 A case handled with homogeneous coordinates ,
6 Space perspectivity ,
7 Perspectivity as a projective transformation ,
8 The case of the trilinear polar ,
9 The case of conjugate triangles.
Related topics :
Menelaus' theorem ,
Homographic relation ,
Projective plane.
- Desargues involution theorem
Desargues theorem on involutions defined on lines through pencils of conics.
- Desargues involution theorem-II
Desargues theorem on involutions defined on lines through pencils of conics. Second case.
- Desargues involution theorem-III
Desargues theorem on involutions defined on lines through pencils of conics. Third case.
- Desargues involution theorem, Complex version
Desargues theorem on involutions defined on lines through pencils of conics. Valid in the complex projective plane.
- Diameter Property 22/06/2006, 3/01/10 euc
Two properties of the diameter of a circle related to angles and products of segments.
- Director circle of an ellipse
An interesting circle from which an ellipse is viewed under a right angle.
- Directrix property created : 15/12/2009 euc
A basic directrix property of conics.
- Division Problem
A fairly general problem and its first instances.
- Diocles cissoid curve
Geometric construction of the cissoid of Diocles.
- Diocles cissoid curve (2)
Cissoid of Diocles in terms of implicit functions.
- Dividing the sides of a triangle
The still unsolved problem of counting all nodal points of a triangle's side-division.
- Division in ratio created : 15/12/2009 euc
To draw a line through a point intersecting two given lines and forming segments of given ratio.
- A distance function
Graph of the distance function between parallel tangents of a circle and an ellipse.
- Domain of convexity
Solution to the problem: Which points produce with their reflections on the sides of a regular hexagon another convex hexagon.
- Domain of convexity for quadrangles
Solution to the problem: Which points produce with their reflections on the sides of a quadrangle another convex quadrangle.
- Droz Farny
Droz Farny theorem on collinearity of points cut on sides of a triangle by two orthogonal lines.
- Duality in projective spaces
The duality principle in projective geometry, interchanging points with lines and joins with intersections.
- Dudeney's dissection
Dudeney's dissection of an equilateral triangle into a square.
- Eight Point Circle
A circle intimately related to an orthodiagonal quadrangle. Starting point for the study of these quadrilaterals.
- Eindvardts Parabolograph
A link-gear mechanism to draw parabolas and their tangents.
- Elations created : 12/12/2007, 28/08/2009 euc
A special kind of perspectivities, concentrating all their fixed points on a line.
- Elation Decomposition created : 12/12/2007, 28/08/2009 euc
Decomposition of an elation in a product of two harmonic perspectivities(homologies) and related questions.
- Eleven Points Conic
A remarkable conic associated to four points and a line in general position.
- Eleven Points Conic II created : 28/08/2009 euc
The eleven points conic generated by harmonic conjugates with respect to two angles.
- Ellipse created : 22/06/2006, updated 11/08/2009 euc
The very basic facts about an ellipse.
- Ellipse from eccentricity created : 15/12/2009 euc
Properties of ellipse deduced from its definition using a focus and a directrix.
- Ellipse as Envelope created : 28/08/2009 euc
A very simple generation of the ellipse as envelope of lines.
- Elliptical caustic
Generated instantly as a geometric locus, through the appropriate tool of EucliDraw.
- Ellipse characterization created : 15/12/2009 euc
Parameterization of ellipse through quadratic rational functions. Determination of geometric characteristics.
- Ellipse Construction
Ellipse Construction through a fairly simple recipe.
- Ellipse Construction (2)
Another ellipse construction through a more simple recipe.
- Ellipse Evolute
The curve enveloping the normals of an ellipse. Geometric construction of curvature centers.
- Ellipse from circles
Ellipse defined as locus of equidistant points from two circles inside one another.
- Ellipse Generation
Ellipse generation through a simple mechanism using two concentric circles.
- Ellipse Generation from circle and point created : 28/08/2009 euc
Ellipse generation by parallel projections of circle-points.
- Enclosing frame of rotated rectangle
Studying the enclosing frame of a rectangle rotating about its center.
- Roulette 3 to 1
Tracing a point on a circle rolling on another circle.
- Equal bisectors property of rectangular hyperbolas
Rectangular hyperbolas as loci of vertices of triangles having two equal bisectors (internal and external).
- Equal circles at vertices
Some properties of the figure resulting by constructing equal circles at the vertices of a triangle.
- Equal circles conic
Definition and first properties of conics defined by taking equal circles at the vertices of a triangle.
- Equilateral and Stewart created : 02/02/2011
A simple application of Stewart's theorem to the equilateral triangle.
- Minimization problem from Euclid
Minimization problem for the area of a triangle originated from Euclid.
- Euler Circle
Some elementary properties and a connexion to the DeLongshampes point.
- Four Euler circles created : 22/06/2006, updated : 31/01/2008 euc
The Euler circles of the four "partial" triangles formed from the vertices of a quadrilateral.
- Euler Circle Property
A property used to animate the rectangular hyperbolas circumscribed on a triangle.
- Euler Conics
The two dual triangle conics related to the Euler line.
- Euler's relation between in/circum- radius created : 10/10/2008 euc
The simple relation between the inradius and circumradius of a triangle.
- Eye curve 5/2/2010 euc
A bicircular quartic generated through a simple geometric locus which is also the inverse of a related ellipse.
- Fagnano's problem
Solution of Fagnano's problem based on reflexions on the sides of a triangle.
- Fagnano's problem II
Another solution of Fagnano's problem based on reflexions on the sides of a triangle.
- Fermat's triangle-point 05/02/2010, 02/02/2011
Fermat's point of a triangle, minimizing the sum of distances from its vertices. Its trilinear coordinates derivation.
- Fermat_Double
A remarkable phenomenon on repeating the construction of equilaterals on the sides of a triangle.
- Fibonacci
A visual proof of an area property of the sequence of Fibonacci integers.
- A five bar linkage
A five bar linkage animated through motor-objects of EucliDraw.
- Locus of focal points of ellipses created : 2/02/2011
A special family of ellipses and the locus of their focal points
- Focal points on an ellipse
An interesting property of the points of an ellipse.
- Four equal circles created : 12/02/2008 euc
Three equal circles centered on a fourth equal to the three have their second intersection points on an equal fifth circle.
- Four-points
Definition and some elementary projective properties of four-points.
- Four-points circumscriptible
Four-points inscribed in circles. Elementary properties.
- Conics through 4 pts and tangent to a line
Given four pts and a line in general position, there are no or two conics passing/tangent to the line.
- Conics through 4 pts and tangent to a line-II
Given four pts in general position, and a tangent passing through one of them.
- Four tangent circles
An interesting point and four tangent circles naturally constructed in a cyclic quadrilateral.
- Fregier created : 22/06/2006, updated : 30/08/2009 euc
Fregier's theorem on conic-involutions, defined by pairs of orthogonal lines through a fixed point of the conic.
- Fregier Involutive homographies
The general behaviour of involutive homographies, preserving a conic.
- Products of involutive homographies
How to find geometrically the composition of involutive homographies, preserving a given conic.
- Fregier Polar
The polar line of the Fregier involution by orthogonals and its practical construction.
- French trio
A combination of three famous theorems to explore the structure of symmetric hexagons.
- Fundamental invariants of the triangle
[New pdf-version] 27/04/2010, 6/1/2019, 23/02/2021
Contents:
1 Fundamental invariants of triangles ,
2 Some remarkable identities ,
3 Generalizing to 3rd degree symmetric functions ,
4 Some 4th degree symmetric functions ,
5 Relations involving the tritangent circles ,
6 The cubic equation satisfied by {a,b,c} ,
7 Blundon’s inequalities ,
8 The GIO triangle ,
9 The orthocentroidal circle ,
10 Euler’s construction problem.
Related topics : Barycentric coordinates.
Conway triangle notation.
- Gauss Bodenmiller theorem
The circles having diameters the diagonals of a complete quadrilateral are coaxal.
- Gear Curve
Parametric form to generate a gear disc with variable radius, teeth height and number of teeth.
- Geometric Series
Proof of the convergence of a geometric series by inspecting triangle division.
- Gergone Universality
The universal property of Gergonne's point of a triangle.
- GIO-Construction of a triangle created : 02/02/2011
Around the construction of a triangle from its elments G(barycenter), I(incenter) and O(circumcenter).
- Glide-Reflexion created : 22/06/2006, updated : 31/01/2008 euc
Definition of glide-reflexion and study of the composition of three reflexions on the respective sides of a triangle.
- Good Parametrizations of Conics
The most "natural" parametrizations of conics, generalizing stereographic projection and establishing homeomomorphism with the projective line.
- Inverse of a good parametrization
The inverse of a good parametrization i.e. rational parametrizations of conics.
- Graphical solution created : 15/12/2009 euc
Graphical solution of the quadratic equation and its generalizations.
- Graphical solution of the cubic created : 15/12/2009 euc
Graphical solution of the cubic equation.
- Graphical solution of the quartic created : 15/12/2009 euc
Graphical solution of the quartic equation.
- Tangent at a point of a function-graph
Tangent at a point of a function-graph and illustration of the dynamic-magnification capability of EucliDraw.
- Graph of a function as a geometric locus
Example of use of the tool [Length-Gauge] of EucliDraw.
- Harmonic Conjugate points 05/05/2008, 05/2/2010 euc
Definition of this basic relation for a quadruple of points on a line. A basic figure connected with them.
- Harmonic pencil of four lines 22/06/2006, updated : 31/01/2008, 05/05/2008 euc
Definition, basic properties and some implication of this basic figure of a quadruple of concurring lines.
- Harmonic pencil problem 05/05/2008, updated : 14/12/2009 euc
A problem easy to solve using properties of harmonic pencils.
- Harmonic common conjugates created : 05/05/2008 euc
The pair of points simultaneously harmonic conjugate to two other pairs.
- Harmonic Perspectivity created : 05/01/10, updated : 10/01/12
Illustration of one of the simplest ways to define a projectivity of the plane.
- Harmonic Quadrangle created : 22/06/2006, updated : 30/08/2009 euc
Cyclic quadrangles such that their vertices, considered as complexes, satisfy (ABCD)=-1.
- Hart lemma
Hart's lemma towards a proof of the famous "Great Poncelet Theorem".
- Hart Linkage
Hart's exact straight-line mechanism, realized through motor-objects of EucliDraw.
- Hart Exact straight line mechanism
Another exact straight-line mechanism, due to Hart. Its geometric background.
- Hexadivision
Dividing the perimeter of a triangle through equidistant points.
- Hexadivision Symmetric
The equilateral hexagon of minimum area inscribed in a triangle.
- Hexagon equilateral inscribed
The center of the equilateral hexagon of minimum area inscribed in a triangle.
- Hexagons symmetric repeated
A sequence of symmetric hexagons generated by a symmetric one.
- Hippopede of Proclus 5/2/2010 euc
A bicircular quartic appearing when studying the chords of a circle through a point.
- Hippopede of Proclus II 5/2/2010 euc
A generation of the previous quartic as envelope of circles passing through a point.
- Hippopede Generalization 5/2/2010 euc
A picture of a curve generalizing a way of generation of the Hippopede of Proclus.
- Homofocal conics
Orthogonality of an ellipse and a hyperbola with common focal points.
- Homographic relation
[New pdf-version] 22/06/2006, 31/01/2008, 26/06/2019
Contents:
1 Homographic Relation ,
2 The group PGL(2,R) ,
3 Fixed points of homographic relations ,
4 Fixed points, the case of involution ,
5 Limit points of homographies ,
6 Homographic relations represent projectivities ,
7 Orthogonality for pencils in involution ,
8 Involutions and pencils of circles ,
9 Homography by polars ,
10 General pencil intersecting a line ,
11 Homographic transformations between lines ,&nbps;
12 Applications to triangle construction problems ,
13 Good parameterizations of pencils and conics ,
14 Homography between a conic and a line ,
15 Homography between two conics,
16 General pencil intersecting a circle ,
17 Three points and a circle through one of them.
Related topics :
Cross ratio.
Projective Line.
- Homography Axis
Study of the homography axis of a homography preserving a conic. Another viewpoint for Pascal's axis.
- Conic-homography 3+3
Behaviour of a conic-homography defined by three points and their images.
- Homothetic Conics
Some questions related to two homothetic conics.
- Homotheties composition created : 02/02/2011
Compositions of homotheties. The basic facts and some applications.
- Homothety property
A trivial property for the intersection-points of two families of parallel lines.
- Hyperbola created : 22/06/2006, updated : 30/08/2009 euc
Basic facts about hyperbolas.
- Hyperbola from middles created : 02/02/2011
A property of hyperbolas related to lines tangent to their auxiliary circle.
- Hyperbola as envelope of lines created : 28/08/2009 euc
Hyperbola generation by parallel projections of circle-points.
- Hyperbola as envelope of lines II created : 28/08/2009 euc
Hyperbola generation by parallel projections of circle-points, generalization by homotheties.
- Hyperbola as a graph created : 05/05/2008 euc
Representing a hyperbola as the graph of a function.
- Hyperbola Asymptotics created : 22/06/2006, updated : 31/01/2008 euc
Basic facts about asymptotic lines of hyperbolas.
- Hyperbola Asymptotics property
Some more facts about asymptotic lines of hyperbolas.
- Hyperbola through its asymptote 5/2/2010 euc
Constructing the hyperbola passing through the vertices of a trapezium and having one given direction for asymptote.
- Hyperbola characterization created : 15/12/2009 euc
Parameterization of hyperbola through quadratic rational functions. Determination of geometric characteristics.
- Hyperbola Construction
Using a typical property of hyperbolas to construct them.
- Hyperbola through equal segments created : 05/05/2008 euc
Generating a hyperbola by tracing equal segments.
- Hyperbola From Ellipse
Creating a hyperbola as image of an ellipse under a projectivity.
- Hyperbola From Rectangular
Derivation of the general hyperbola from the standard one (graph of y=1/x) through an affinity.
- Hyperbola Generation
Creating a hyperbola as image of a circle under a projectivity.
- Hyperbola Generation II
Creating a hyperbola as image of a circle under a projectivity. A second variant.
- Hyperbola generated by line intersections created : 28/08/2009, updated : 12/12/2009 euc
Hyperbola generation by intersecting radii of moving points on circles.
- Hyperbola with given asymptotics created : 28/08/2009, updated : 12/12/09 euc
Hyperbola with given asymptotics and passing through one given point.
- Hyperbola Property
A property of hyperbolas related to its asymptotic lines.
- Hyperbola as locus of points created : 28/08/2009 euc
Hyperbola generation as locus of points intersecting segments ending on fixed lines in fixed ratio.
- Hyperbola Property with middles
A property of hyperbolas as locus of middles of segments through fixed point intersepted by two lines.
- Hyperbola Property with parallels created : 22/06/2006, updated : 05/05/2008 euc
Creating a hyperbola by varying to parallel lines. A subject with interesting applications.
- Hyperbola as locus of points created : 22/06/2006, updated : 05/05/2008 euc
Hyperbola as locus of points on segments through fixed point A and intersecting these segments in fixed ratio.
- Hyperbola Rectangular
Basic properties of rectangular hyperbolas, having orthogonal asymptotic lines.
- Hyperbola related to equilateral created : 11/08/2009, updated : 12/12/09 euc
A special hyperbola generated by secants and tangents at points symmetric with respect to a diameter.
- Hyperbola with respect to its asymptotics created : 22/06/2006, updated : 31/01/2008 euc
The equation of the hyperbola with respect to its asymptotic lines. The family generated by the hyperbola and its asymptotes.
- Hypocycloid
A curve generated by a fixed point on a circle rolling inside another circle.
- Hypotrochoid
A curve generated by a point rigidly attached to a disc rolling inside circle.
- Implicit function
A curve whose points are determined through an equation.
- Incircle of a triangle 02/02/2011
Basic facts on the incircle and Excircles of a triangle.
- Incircle dual
A remarkable ellipse, representing the dual of the incircle relative to a triangle.
- Incircle in trilinears 5/2/2010 euc
Calculation of the incircle and the excircles of a triangle in trilinear (and/or barycentric) coordinates.
- Incircle tangents created : 22/06/2006, updated : 05/05/2008 euc
Generating the tangents of the incircle of the equilateral triangle through trilinear polars.
- Incircle tangents II
A trivial remark on the previous topic, useful in other topics.
- Inclined Nodes
A remarkable curve, generated geometrically and having a simple equation.
- Inconics in trilinears 5/2/2010 euc
The form of equations in trilinear coordinates of conics tangent to all three sides of a triangle. The role of perspector. The duality to circumconics.
- Inconics tangents created : 22/06/2006, updated : 05/05/2008 euc
Generation of an inconic of a triangle as envelope of trilinear polars.
- The triangle of the radical axes of three circles
Resembles to the tritangent circles of a triangle ... but not exactly.
- Conics inscribed in triangles created : 22/06/2006, updated : 05/05/2008 euc
Conics inscribed in triangles generated as envelopes of lines.
- Inscribed Squares 22/06/2006, 02/02/2011
The 6 inscribed squares of a generic quadrangle.
- Inscriptible Construction
Construct a quadrangle inscriptible in a circle through its side-lengths.
- Intersection of a conic with a line created : 02/01/12
Construct the intersection points of a conic given by five points and a line.
- Inversion [New pdf-version] 01/10/2007, 15/06/2019
Contents:
1 Inversion on a circle ,
2 Inverses of lines and circles ,
3 Metric relations ,
4 Ptolemy’s theorem ,
5 Inversion is a conformal map ,
6 Inversion preserves the cross-ratio ,
7 Inversion and pencils of circles ,
8 Inversion interchanging two circles ,
9 Inverseness is preserved by inversions ,
10 Inverting to equal circles ,
11 Composition of two inversions ,
12 Composition of three inversions ,
13 Inversion action on an invariant circle ,
14 Compositions of two inversions and homographies ,
15 Anti-inversion ,
16 Three pairwise orthogonal circles ,
17 Inversions and complex numbers.
Related topics :
Circle pencils,
Cross Ratio,
Homographic relation,
- Inversion generalization created : 15/12/2009 euc
Generalization of the inversion in the projective plane, with respect to some conic.
- Inversion Property
A property of the inversion with respect to a circle used to prove Archimedes' Circles theorem.
- Inverting on a family of circles
The envelope of circles created by inverting a circle with respect to members of a family of circles.
- Homographic involutions and circle pencils
A simple and nice interpretation of a homographic involution through a circle pencil. Connection of pencil-characteristics with fixed points of involution.
- Homographic involutions and circle pencils II
A procedure which, in some sense, is inverse to the one described in the previous reference.
- Homographic involutions and circle pencils III
A slightly more general than the previous precedure of definition of an involutive homography.
- Basic example of involutive homography created : 22/06/2006, updated : 31/01/2008 euc
The involutive homography defined on a line through the intersections with the sides of a complete quadrilateral.
- Basic example of involutive homography-II
The involutive homography defined on a line through the intersections with the sides of a complete quadrilateral. Determination of fixed points.
- Basic example of involutive homography-III
The involutive homography defined on a line through the intersections with the sides of a complete quadrilateral. Just another figure.
- Product of homographic involutions
The product of two homographic involutions and its characteristics expressed in terms of the data of the factors.
- Product of two homographic involutions created : 22/06/2006, updated : 31/08/2009 euc
The general case of a product of two harmonic perspectivities of the projective plane.
- Involutive homography created : 22/06/2006, updated : 31/08/2009 euc
Homography on a conic which is inverse to itself. The basic properties.
- Isogonal on the bisector 5/2/2010 euc
A study of the isogonality with respect to a bisector of a triangle.
- Isogonal conjugation
How to define the transformation of isogonal conjugation with respect to a triangle.
- Isogonal conjugation (2)
Application of the isogonal conjugation on the common tangents of three circles.
- Isogonal as quadratic
The isogonal transformation as a quadratic transformation defined by a family of conics.
- Isogonal generalized created : 22/06/2006, updated : 31/01/2008, 05/05/2008 euc
The general "isogonal" transformation, associated to a triangle and a point, as a quadratic transformation defined by a family of conics.
- Isogonal of circumcircle created : 05/05/2008 euc
Application of the isogonal conjugation to points on the circumcircle of the triangle of reference.
- Isogonal of parabola created : 05/05/2008 euc
Lines whose isogonals with respect to the triangle of reference are circumscribing parabolas.
- Isogonal of parabola II created : 05/05/2008 euc
Lines whose isogonals with respect to the triangle of reference are circumscribing parabolas. Further properties.
- Isometries or Congruences of the plane [New pdf-version] 22/11/2009, 15/10/2019
Contents:
1 Transformations of the plane,
2 Isometries, general properties,
3 Reflections or mirrorings,
4 Translations,
5 Rotations,
6 Congruence,
7 Some compositions of isometries.
- Products of isometries (2)
Determination of the composition of rotations about the vertices of a quadrangle + a translation.
- Isosceles intersection
A characteristic property of the circumcenter of an isosceles triangle.
- Isosceles property
A property of the isosceles representing a clue on the topic of trilinear polars.
- Isotomic as quadratic
Representation of the isotomic conjugation as a quadratic transformation relative to a family of conics.
- Isotomic Chart created : 05/05/2008 euc
Diving into the incidence relations of trilinear polars of isotomic points with respect to a triangle and related lines and conics.
- Isotomic Conic of a Line created : 22/06/2006, updated : 05/05/2008, 10/10/2008 euc
A recipe to obtain conics from lines using trilinear coordinates and isotomic conjugation.
- Isotomic General created : 05/05/2008 euc
Some properties of the generalized isotomic transformations with respect to a triangle.
- Isotomic of circle created : 05/05/2008 euc
The line whose isotomic with respect to a triangle is its circumcircle.
- Isotomy on median created : 22/06/2006, updated : 31/01/2008 euc
Determination of the isotomic transformation w.r. to a median in trilinears.
- Isotomic Chart created : 05/05/2008 euc
Relations between the trilinear polars and other remarkable lines and their isotomic conjugates.
- Iterative
What is an iterative procedure for the determination of roots of equations.
- Iterative applied to cos(x)
Explanation of a curious phenomenon when playing with a calculator supporting the cos(x) function.
- Jerabek hyperbola
The Jerabek rectangular hyperbola of a triangle.
- Kepler Problem
A problem of Kepler asking the construction of the maximal barel inscribed in a sphere.
- Kiepert hyperbola
The Kiepert rectangular hyperbola of a triangle.
- Koch Snowflake curve
A fractal curve obtained by a simple construction, starting with an equilateral triangle.
- Koch Snowflake generalized curve
A slightly more general than the previous one fractal curve. Construction based on arbitrary triangle.
- Koch Snowflake generalized curve-2
A slightly more general than the previous one fractal curve. Construction based on arbitrary rectangle.
- The Koch Snowflake general curve
Illustration of a fairly general procedure to construct Koch curves, out of arbitrary polygons and arbitrary broken lines.
- Yet another Koch fractal curve
A more efficient general procedure to construct Koch fractal curves, out of arbitrary polygons and arbitrary broken lines.
- Laisant Trisector
A mechanical device to trisect angles.
- Lahire's triangle construction
Solving the problem of triangle ABC construction, from the angle A, the sum b+c and the altitude from A. created : 25/03/2016
- Lemniscatoids
Four-bar crossed-crank linkage for drawing Lemniscatoids.
- Levy Tapestry
A fractal producing a nice tapestry motif. Using a custom tool of EucliDraw.
- Limacon created : 22/06/2006, updated : 31/01/2008 euc
First properties of Pascal's limacon.
- Limacon II created : 31/01/2008 euc
Further properties of Pascal's limacon.
- Lines created : 15/12/2009 euc
Basic facts about lines and their representations through coordinates.
- Line or tangential coordinates 5/2/2010 euc
An introduction to line or tangential coordinates which parameterize lines like usual coordinates parameterize points.
- Line pencils created : 15/12/2009 euc
Pencils of lines passing through a point. Basic properties and aspects.
- Line Homography
Homography between the points of two lines.
- Linear combination of polygons
Polygon moving while remaining similar to itself and two vertices glide on two fixed lines.
- Line Homography axis
A fundamental property of homographies between points of two lines.
- Line Homography axis II
The true story about the line homography axis. Connexion with Chasles-Steiner conics.
- Line equation in trilinear coordinates
Line equation in trilinear coordinates. The dependence on coefficients.
- Line locus from ratio created : 05/05/2008 euc
A simple line locus related to the constancy of a certain ratio.
- Line Similarity axis
Just a special case of the previous subject.
- Lines of segments
A natural variation of a segment having its end points on two lines.
- Lines of segments II
Connexion of the previous topic with a parabola and relations to a generalized Thales theorem.
- Link Gear Dwell
Realization of a link gear mechanism through motor-objects of EucliDraw.
- Link Point Locus
The locus of a point on a link, whose end-points move on two circles.
- Lissajous curves
Mechanical generation of the Lissajous curves through motor-objects of EucliDraw.
- Locus Concentric
A simple geometric locus, related to two intersecting circles.
- Maclaurin
The Maclaurin way of constructing conics through intersections of varying line-pairs.
- Maclaurin, Chasles, Steiner created : 02/02/2011
Some further discussion on the generation of conics through the Maclaurin - Chasles - Steiner method.
- Maclaurin dual created : 15/12/2009 euc
Dual of the Maclaurin theorem on the generation of conics as envelopes of lines.
- Maclaurin like generation of conics created : 10/10/2008 euc
A way to generate conics somewhat similar to Maclaurin's.
- Maclaurin like generation of conics II created : 10/10/2008 euc
A way to generate conics somewhat similar to Maclaurin's, necessity of a collinearity.
- Malfatti created : 22/06/2006, updated : 31/08/2009, 11/01/2013 euc
Hart's completion of Steiner's solution of Malfatti's problem.
- Mathot
The Monge/Mathot point (or anticenter) of a circular quadrangle.
- Mathot (2)
The Mathot point of a circular quadrangle. Application of the theorem (proof).
- Mathot (3)
The Mathot point of a circular quadrangle. The full image of the previous theorem.
- Maximal cyclic quadrangle 5/2/2010 euc
On the maximal quadrangle inscribed in a circle whose diagonals pass through a fixed point.
- Maximal Ellipse
On the maximal, in area, ellipse inscribed in a square. The obvious solution.
- Maximal Rectangle
On the maximal, in area, rectangle inscribed in a circle.
- Maximal Rectangle in Ellipse
On the maximal, in area, rectangle inscribed in an ellipse. Its relation to the axes.
- Maximal Polygons in Ellipse
On the maximal, in area, N-sided polygons inscribed in an ellipse.
- Maximal Segment
Maximizing a segment joining two points on two circles.
- Maximal Triangles in Ellipse
On the maximal, in area, triangles inscribed in an ellipse. Some elementary properties.
- Maximal Triangles Properties
On the maximal, in area, triangles inscribed in an ellipse. Additional properties.
- Maximum area
Maximum area property of inscribed quadrangles. Using shapes and combining them with graphs of functions.
- On the medial line of a side of a triangle
Study of some properties of the medial line of a side of a triangle.
- Medial Parabola
The parabola, wich is tangent to the three sides and the medial line of a side of a triangle.
- Median property
A property of the medians related to some rectangular hyperbolas associated with a triangle.
- Median triangle created : 11/08/2009 euc
The median triangle of a triangle ABC and its relation to Vecten configuration and Brocard angle of ABC.
- Menelaus' theorem [New pdf-version]
22/06/2006, 02/02/2011, 22/06/2019
Contents:
1 Menelaus’ theorem 1 ,
2 Menelaus applications 2 ,
3 Menelaus projective 3 ,
4 Menelaus from Ceva 4 ,
5 Applications of Menelaus’ theorem II.
Related topics :
Cross Ratio,
- Mid Circle
The circle defining an inversion which interchanges a pair of circles.
- Mid Circle-2
The circle defining an inversion which interchanges a pair of circles. The case of two tangent circles.
- Mid Circle-3
The circle defining an inversion which interchanges a pair of circles. The case of a circle inside another.
- Mid Circles
The circles defining an inversion which interchanges a pair of circles. The case of two intersecting circles.
- Mid Circles-2
The circles defining an inversion which interchanges a pair of circles. The case of a circle and an intersecting line.
- Minimal Ellipse
On the minimal, in area, ellipse enclosing a square. The obvious solution.
- Minkowski Sausage
A fractal resulting by repeatedly replacing the sides of a polygon with a broken line.
- Mid-Circle preservetion by inversions
The preservetion of the mid-circle of two circles by inversions.
- Minimal Ellipse of a parallelogram
On the minimal, in area, ellipse enclosing a parallelogram. Reducing to the above obvious solution.
- Miquel
Miquel's theorem on four intersecting circles.
- Miquel Dual created : 31/01/2008 euc
A sort of dual of Miquel's theorem leading to triangles pivoting about a point.
- Miquel (2)
Miquel's theorem on four intersecting circles. A second instance.
- Miquel Pentagon
Miquel's theorem on the pentagram created from an arbitrary pentagon.
- Miquel Point created : 22/06/2006, updated : 31/01/2008 euc
Miquel's pivot point, Miquel's theorem on four intersecting lines and consequences for parabolas.
- Miquel Triangle
Similarity of some triangles in the Miquel configuration.
- Mittenpunkt X(9)
Some properties of the triangle center X(9) and a relation to triangle center X(84).
- Mixtilinear circles created : 31/01/2008 euc
Definition and basic property of mixtilinear circles of a triangle.
- Moebius involution created : 15/12/2009 euc
An example of a Moebius involution related to the ratio of two segments on a line.
- Moebius transformations created : 11/08/2009 euc
Definition and basic properties of Moebius transformations.
- Moebius diagram of a triangle created : 11/08/2009 euc
Definition and basic properties of the Moebius diagram of a triangle configuring classes of iso-brocardian triangles.
- Moebius transformation preserving a circle
Properties of such a transformation, related to circle-pencils.
- Moon Motion
Simplified absolute motion of the moon. Considering movement in elliptic trajectories.
- Moving similar polygons
Moving a polygon similar to itself about a fixed vertex, another vertex gliding on a line.
- Nagel point of the triangle [New pdf-version] 06/07/2010, 04/02/2019
Contents:
1 Nagel point of the triangle ,
2 Barycentric coordinates of the Nagel point ,
3 The Nagel line of the triangle ,
4 Alternative construction of the Nagel point ,
5 Other Nagel-like points ,
6 Connection with the de Longchamps point.
Related topics : Barycentric coordinates,
De Longchamps point of the triangle.
- Napoleon triangle(s)
Definition of the Napoleon triangle(s) of a triangle.
- Nephroid
A simple geometric construction of the Nephroid.
- Nested Polygons
How to construct self-repeating schemes using EucliDraw's scheme-sockets tools.
- Nested Rotated Polygons
How to construct self-repeating schemes using EucliDraw's scheme-sockets tools. A variation on the previous.
- Newton's way to generate conics 5/2/2010 euc
Study of the way Newton generated conics by rotating two angles and taking intersections of their legs.
- Newton Line of a quadrangle created : 22/06/2006, updated : 05/05/2008 euc
A property studied by Newton: The middles of the three diagonals of a complete quadrangle are collinear.
- Newton line for circumscriptible
The Newton line of a circumscriptible quadrilateral passes through the incenter.
- Newton Iterative
The iterative procedure of Newton to determine roots of equations.
- Newton Iterative II
The iterative procedure of Newton to determine roots of equations. A second example.
- Newton line property created : 11/08/2009 euc
Two points on the Newton line of a quadrilateral related to the three middles of its diagonals.
- Conchoid of Nicomedes
Definition and construction of the Conchoid of Nicomedes.
- Conchoid of Nicomedes (2)
Conchoid of Nicomedes defined by an implicit function.
- Nine Points Conic
The conic passing through the 6 middles of sides of a complete quadrilateral.
- Closed even sided polygons with given middles
Examples of non-symmetric closed polygons with prescribed middles of their sides.
- Object Lattice
Constructing latices of objects through user-tools of EucliDraw.
- Olympiad problem
A problem from the olympiad contests, where the similarity of triangles plays a particular role.
- Orthic triangle created : 22/06/2006, updated : 10/10/2008 euc
The triangle of the traces of the altitudes on the sides of a triangle.
- Orthic Axis
The orthic axis of a triangle, related to the orthic triangle and the Euler line.
- Orthocenter created : 22/06/2006, updated : 10/10/2008 euc
Vector properties of the orthocenter of a triangle.
- Orthocenter2
Further properties of the orthocenter of a triangle.
- Orthocenter Projections
Projecting the orthocenter on the line joininig a side-middle with the center of Euler's circle.
- Orthocentric Anti-Inversion
An anti-inversion related to the orthocenter of a triangle.
- Orthocycle
A circle intimately related to a cyclic quadrangle.
- Orthocycle and bicentrics
A perspective using the orthocycle to study the bicentric quadrilaterals inscribed in a circle.
- Orthocyclic Characterization
A characterization of the cyclic quadrilaterals using their orthocycle.
- Orthodiagonal quadrangle
Some basic properties of orthodiagonal quadrangles going back to Steiner.
- Orthodiagonal quadrangle-2
Illustrating some points discussed in the previous file.
- Orthodiagonal quadrangle-3
Some further properties of these remarkable quadrilaterals.
- Orthodiagonal quadrangle-4
And here the hints or proofs of their properties.
- Orthodiagonal from Cyclic
A natural map associating to every cyclic quadrilateral a orthodiagonal one.
- Minimal orthodiagonal quadrangle
Given the length of the diagonals, find the minimal in perimeter orthodiagonal quadrangle.
- Orthogonal Diagonals
Various properties of the orthodiagonal quadrangle, resulting by reflecting a point on the sides of rectangle.
- Orthogonal pencils
A remark concerning the powers with respect to the circles of circle pencil.
- Orthopolar triangles created : 10/10/2008 euc
Triangles related through their corresponding deltoids.
- Orthopole of a line created : 10/10/2008 euc
The orthopole point of a line with respect to a triangle.
- Orthopole of a line II created : 10/10/2008 euc
The orthopole point of a line with respect to a triangle. Additional properties.
- Triangle inscribed in a Rectangular Hyperbola created : 22/06/2006, updated : 31/01/2008 euc
The very basic properties of a triangle inscribed in a Rectangular Hyperbola.
- Oscillating circle of a conic
A simple recipe to locate the oscillating circle at a point of a conic.
- Pappus
The theorem of Pappus, generalizing that of Pythagoras, and giving the inspiration for the EucliDraw-logo.
- Pappus application created : 22/4/2013
An application of Pappus theorem leading, after a few steps, to an autopolar triangle w.r. to the Euler circle.
- Pappus Lines
The theorem of Pappus, on the hexagons with vertices on two lines.
- Pappus Lines II
The theorem of Pappus, a special case to which reduces the general one.
- Pappus Self Dual
The theorem of Pappus, and its characteristic to be self-dual.
- Pappus Triangle Construction created : 22/06/2006, updated : 12/12/2009 euc
Constructing a triangle from the three elements (A, a, b_{A}) and related problems.
- Parabola created : 22/06/2006, updated : 11/08/2009 euc
Definition and the very basic properties of the parabola.
- Parabola Problems created : 22/06/2006, updated : 28/02/2013
Discussion of some problems related to parabolas, together with a contribution by Mehmet Kilic.
- Parabola properties created : 12/12/2009 euc
Further properties of the parabola.
- Parabola Area
The way Archimedes calculated the area of a sector of a parabola.
- Parabola by similar triangles
A parabola connected to an elementary property of circles.
- Parabola through centers created : 31/01/2008 euc
Parabola as a locus of centers of circles which are simultaneously tangent to a given line and a given circle.
- Parabola Chords created : 22/06/2006, updated : 31/01/2008 euc
The basic properties of the chords of a parabola.
- Parabola circumscribing trapezium created : 11/08/2009 euc
The parabola passing through the four vertices of a trapezium. Basic properties.
- Parabola from equal segments created : 05/05/2008 euc
Parabola generated by certain equal segments.
- Parabola from projections
A parabola generated by diagonals of quadrangles defined through projections.
- Parabola Homographies
Homographies preserving a parabola and fixing a point of it.
- Parabola inscribed in a triangle created : 05/05/2008 euc
Study of the parabola inscribed in a triangle.
- Parabola inscribed in a quadrangle created : 05/05/2008 euc
Study of the parabola inscribed in a quadrangle.
- Parabola as envelope of angle-sides created : 31/01/2008 euc
Parabola as envelope of an angle-side whose other sides passes through a fixed point and the vertex moves on a line.
- Parabola Parameter
Determination of the parameter of a parabola from the coefficient of the standard equation.
- Parabola property
A property of parabolas concerning the cross ratio, defined by five tangents.
- Parabola to oblique axes
Determination of the focus and directrix of a parabola defined by an equation in oblique axes.
- Parabola Symmetries
Some affine symmetries of the parabola and related properties.
- Parabola inscribed in equilateral created : 05/05/2008 euc
Equilateral triangles tangent to a given parabola.
- Parabola trapezium
Two additional points on a parabola, defined by two points and its tangents there.
- Parallelogram division
How to divide a parallelogram in four equal, in area, parts, through two lines.
- Parallelograms inscribed in ellipse
General properties of parallelograms inscribed in an ellipse. The maximal in area parallelograms.
- Parallelograms inscribed in ellipse-2
Parallelograms inscribed in an ellipse and having constant area E.
- Parallelograms inscribed in ellipse-3
Parallelograms inscribed in an ellipse, having constant area E and maximal/minimal perimeter.
- Parallels Medians
A case of constancy of direction of a variable line.
- Parallels to Ellipse
Parallels to Ellipse produced through the tool of geometric-loci of EucliDraw.
- Parametric cubic
Construction of the parametric cubic, whose coefficients are four given points.
- Pascal created : 22/06/2006, updated : 11/01/2006, 12/12/2009 euc
Pascal's theorem on hexagons inscribed in conics.
- Pascal (2) 22/06/2006, 31/01/2008, 31/08/2009, 02/02/10 euc
Pascal's theorem on hexagons inscribed in conics. A simple nice proof and a special case.
- Pascal-Brianchon duality
Pascal-Brianchon duality based on polar reciprocity.
- Pascal and Euler
A point on the Euler circle of a triangle controlling some circumconics of the triangle.
- Pascal and its inverse
A question on circumconics of a triangle where the theorem of Pascal as well as its inverse apply.
- Pascal on quadrangles created : 22/06/2006, updated : 05/05/2008 euc
A special case of Pascal's theorem concerning quadrangles.
- Pascal on triangles created : 22/06/2006, updated : 05/05/2008 euc
A special case of Pascal's theorem concerning triangles.
- Pascal Poncelet Brianchon Meeting
A figure illustrating three major theorems of projective geometry.
- Peaucellier's Inversor
Well known mechanism to transform circular to rectilinear motion.
- Pedal triangle of a point relative to a triangle 22/06/2006, 31/08/2009, 02/02/10, 25/03/21, 30/04/21, 05/05/21
Basic properties of the triangle with vertices the projections of a point on the sides of another triangle.
Contents:
1 Definition and first properties ,
2 The pedals of points in various regions ,
3 How to inscribe, 12 pivots ,
4 Invariance under inversions ,
5 Invariance under Apollonian inversions ,
6 Invariance under the circumcircle inversion ,
7 Reduction to the equilateral ,
8 The dual viewpoint ,
9 In‑pivots and circum‑pivots ,
10 Area of pedal ,
11 Formularium ,
12 The third pedal ,
13 Pedal, cevian and circumcevian triangles ,
14 Darboux cubic ,
15 Pedals of isogonal points ,
16 More on pedals of isogonal points .
Related topics :
Apollonian circles of a segment,
Apollonian circles and isodynamic points of the triangle,
Barycentric coordinates,
Cross Ratio,
Inversion transformation,
BProjective line,
Projectivities,
- Pedal problem
A concurrence problem related to the pedal triangle of a point wiht respect to a triangle.
- Pedals of ellipses
Generation of various pedals through the tool of geometric-loci of EucliDraw.
- Pedal Polygons
Polygons resulting by projecting a point on the sides of a polygon. Study of their area.
- Pentadivision
Construct an equilateral pentagon inscribed in a triangle.
- Pentadivision Degenerate
Construct an equilateral degenerated pentagon inscribed in a triangle.
- Pentagon Equilateral
Construct an equilateral pentagon from some minimal data.
- Perrolatz Inversor
A linkage realizing an inversor with respect to a circle.
- Perspective triangles created : 02/02/2011
A case of perspective triangles studied in projective coordinates.
- Perspectivity to the Artzt parabola created : 10/10/2008 euc
On the perspective images of the Artzt parabola.
- Perspectivity Through Matrix
A special kind of perspectivity, defined by a matrix.
- Pick
Pick's theorem on measuring the area of a domain through counting of latice-points.
- Philon of Byzantium problem
Philon's problem on the minimal segment through a fixed point, intercepted by two lines.
- Philon's cubic
A cubic curve intimately related to the problem of Philon.
- Cubic equation for Philon's problem
A cubic equation satidied by the tangens of the slope of Philon's line.
- Pirate's treasure
On treasures and pirates that know geometry.
- Piriform Curve
Definition and construction of the Piriform curves.
- Piriforms
Mechanical production of Piriform curves, through linkages.
- Pivots of inscribed triangles created : 22/06/2006, updated : 31/08/2009 euc
Points defined naturally when inscribing triangles into other triangles.
- Points dividing in given ratios
Construct a triangle, whose sides pass through three given point, and satisfies a certain condition.
- Polar created : 22/06/2006, updated : 02/02/2011
The polar line of a point with respect to a circle. Definition and first properties. Generalization to conics.
- Polar construction
A simple recipe to construct the polar of a point with respect to a circle.
- Polarity and cross ratio
Preservation of the cross ratio by a polarity with respect to a circle.
- Polar properties
Some properties of the polar line of a point with respect to a circle.
- Polar properties-2
Some further properties of the polar line of a point with respect to a circle. An overloaded picture.
- Polar on two lines created : 02/02/2011
Definition of the polar line of a point with respect to two lines. The concept of the conjugate polygon with respect to a point.
- Polar And Inversion
Some properties deriving from relations of polars to inversions.
- Polar locus
A remarkable locus of the orthocenter of a triangle directly related to the polar of a point w.r. to a circle.
- Polar Construction
Polar Construction for conics.
- A property of polar
A property of the eccentric angles of intersection points of a polar with the respective conic.
- Pollock
Pollock's configuration.
- Pollock (2)
Pollock's configuration. The statement of the theorem and its proof.
- Polynomial Property
A property of curves defined through graphs of polynomials, generalizing the Newton-Vieta relations of roots.
- Poncelet porism
Poncelet's "great" theorem on polygons inscribed and simultaneously circumscribed about conics.
- Poncelet's angle relations created : 31/01/2008 euc
Poncelet's relations between angles formed by tangents to conics and focal radii.
- Poncelet proof
An outline of an ingenious proof of "Poncelet's great theorem", based on a lemma by Hart.
- Poristic triangles created : 10/10/2008 euc
On one-parameter families of triangles. A simple example.
- Power genearalization 22/06/2006, 31/01/2008, 12/12/09, 02/02/10 euc
Some properties of conics which reduce to properties of the power in the case of circles.
- Power along asymptotic directions 5/2/2010 euc
Specialization of the previous for the case of secants in the direction of asymptotes of a hyperbola.
- Power along axial direction 5/2/2010 euc
Specialization of the previous for the case of secants in the direction of the axis of a parabola.
- Projection Polygon
The polygon of feet of perpendiculars from a point on the sides of another polygon.
- Projection Triangle
The triangle of feet of perpendiculars from a point on the sides of another triangle.
- Quadrangles projective collinearity
A projectivity associated naturally with a general quadrangle.
- Projective Line [New pdf-version] 31/01/2008, 26/10/2018, 26/06/2019, 14/12/21
Contents:
1 Projective line,
2 Homogeneous coordinates,
3 Projective base,
4 Relation between euclidean and projective base,
5 Change of projective bases,
6 Line projectivities or homographies,
7 The fundamental theorem for line projectivities,
8 Homographic relations, Moebius transformations,
9 Homographic relations, the group properties,
10 Line perspectivities,
11 Circle and tangents homography,
12 Cross ratio or anaharmonic ratio,
13 Projectivities defined by cross ratios,
14 Cross ratio in euclidean coordinates.
Related topics :
Cross Ratio,
Homographic relation,
- Projective Plane [New pdf-version] 22/07/2011, 08/07/2019
Contents:
1 Projective plane, the standard model ,
2 Projective base and associated projective coordinates ,
3 Projective lines ,
4 Cross ratio ,
5 Trilinear polar and pole ,
6 Projectification of the euclidean plane ,
7 Homogeneous coordinates ,
8 Homogeneous coordinates relation to projective coordinates ,
9 Changing the system of projective coordinates ,
10 Obtaining equations in a projective coordinate system ,
11 Example calculation of the radical axis of two circles ,
12 The circumcircle of ABC and the general circle ,
13 Projective transformations or projectivities ,
14 A proof of Pappus’ theorem ,
15 Presevation of the cross ratio ,
16 Projective conics ,
17 Pole and polar ,
17 Projectivities mapping a circle to a parabola/hyperbola.
Related topics :
Barycentric coordinates,
Ceva's theorem,
Cross Ratio,
Menelaus' theorem,
Projective line,
The quadratic equation in the plane,
- Projectivities fixing vertices
Study of properties of projectivities fixing the vertices of a triangle.
- Projective Rotations
Projectivities conjugate to usual rotations with some applications to triangle geometry.
- Projectivities
Definition and main properties of the transformations prevailing in projective geometry.
- Resolution to perspectivities
Resolution of a general projectivity to perspectivities.
- Pythagora's theorem 11/08/2009, 02/02/10 euc
Pythagora's theorem and some simple consequences.
- Quadrangle construction
Construction of a quadrangle with three equal sides by giving the middles of these sides.
- Quadrangle division in four
How to divide a quadrangle in four equal, in area, pieces, through two lines.
- Quadrangles classified modulo affinities
The moduli space of quadrangles and a representative for each class, modulo affinities.
- Quadrangle 11/08/2009, 02/02/10 euc
Basic intersection properties on sides and diagonals of a generic quadrangle.
- Quadrangle's harmony 5/2/2010 euc
A picture illustrating the properties of a general quadrangle related to harmonicity.
- Quadratic solving 5/2/2010 euc
What are we doing when we solve a quadratic equation?
- The quadratic equation in the plane [New pdf-version] 10/01/2012, 16/02/2019, 09/02/2021, 27/10/2021, 10/03/24
Contents:
1 Introduction ,
2 The allowed coordinate systems ,
3 The transformation of the coefficients ,
4 The invariants ,
5 Product of lines,
6 Degenerate conics ,
7 Proper conics ,
8 Central conics ,
9 Find the center,
10 Find the center, examples ,
11 Axes of central conics,
12 Finding the normal form ,
13 Example calculation of the normal form ,
14 Finding the axes of the conic ,
15 Finding the kind of the conic,
16 Asymptotes ,
17 The angle of the asymptotes ,
18 Rectangular hyperbola ,
19 Asymptotes directly ,
20 Parabolas ,
21 Parabola Examples ,
22 Matrix representation, tangents , secants
23 Conjugate directions
24 Polar and pol ,
25 Quadratic equation classification ,
26 Can you easily find a point on the conic?
27 On the focal points
- Quadratic Transformation
Illustration of the quadratic transformation, defined through a family of conics.
- Quadratic Transformation (2)
Applying quadratic transformations to lines, to produce conics.
- Quadratrix
The curve used to devide angles in a given ratio.
- Quartic
The (affine) symmetries of a curve created by a polynomial of fourth degree.
- Radical Axis of two circles created : 08/08/2007, updated : 31/01/2008 euc
Radical Axis of two circles and a plausible property.
- Radical Axis of two circles II
Radical Axis of two circles. A simple property.
- Ratio of distances from middle
Calculation aid for ratios defined by points on a segment.
- Reciprocal polar conics
A duality based on the pole-polar reciprocity with respect to fixed conic.
- A property of conjugate lines of a rectangular hyperbola
A very simple property of conjugate lines of a rectangular hyperbola.
- Rectangular Hyperbola
A rectangular hyperbola defined by a simple relation.
- Rectangular Hyperbola from a line created : 02/02/2011
A family of rectangular hyperbolas defined by a line in a Cartesian coordinate system.
- Rectangular Hyperbola Circumscribed created : 08/08/2007, updated : 31/01/2008 euc
Construction-Animation of all rectangular hyperbolas circumscribed on a triangle.
- Impossibility to inscribe a rectangular Hyperbola
Illustration of the typical case where it is impossible to inscribe in a quadrangle a rectangular hyperbola.
- Creating parallel chords in rectangular hyperbolas created : 08/08/2007, updated : 31/01/2008 euc
Intersecting a rectangular hyperbola with a circle pencil and creating parallel chords.
- Rectangular hyperbola generation
A very easy way to construct a rectangular hyperbola.
- Rectangular Hyperbolas tangent to four lines
Construction of the (two) hyperbolas tangent to four given lines (when this is possible).
- Rectangular Hyperbola through four points created : 08/08/2007, updated : 31/01/2008 euc
Construction of the rectangular hyperbola passing through four non orthocentric points.
- Rectangular Hyperbola inscribed in a triangle
Construction of the rectangular hyperbolas (when existing) tangent to the three sides of a triangle.
- Recycler
A Moebius transformation recycling the vertices of a triangle.
- Recycler Homographic
A Homographic transformation recycling the vertices of a triangle.
- Reflexion on conics
A conic involution resembling the usual reflexion, but defined on a conic.
- Reflecting On Polygon Sides
The composition of reflection on the sides of a quadrangle.
- Reflecting On Polygon Sides (2)
The composition of reflection on the sides of a quadrangle. The final picture.
- Reflecting segment ends
A segment of fixed length gliding on a side of a triangle and its reflexions on the other sides.
- Reflexions of line
A parabola connected with closed billiard trajectories inside a triangle.
- Regula Falsi
The method of "false position" for the determination of the roots of an equation.
- Rhodonea
Curves similar to flowers with any number of petals.
- Rolling On Parabola
Can a wheel roll on a parabola?
- Roots
Construction of qudratic roots of integers. Using scheme-sockets.
- Roots (2)
Construction of qudratic roots of integers. Using scheme-sockets. A second method.
- Rotating a right angle into a circle 5/2/2010 euc
A study of the rotation of a right angle inside a circle. A related generation of an ellipse.
- Rotating the sides of a triangle
Some figures resulting by rotating the sides of a triangle about their middles.
- Rotating the sides of a triangle (2)
Further study of figures resulting by rotating the sides of a triangle about their middles.
- Rotating the sides of a triangle (3)
Relating Brocard Geometry with figures resulting by rotating the sides of a triangle about their middles.
- Rotating Triangle
How to construct a triangle rotating in its circumcircle.
- Rotating Triangle similar
Some properties related to a triangle, rotating in its circumcircle.
- Rotation at point A by an angle w 11/08/2009, 02/02/10 euc
Analytic description of the transformation and a simple application.
- Rotation product odd
Calculating the composition of rotations on the vertices of an odd-sided polygon.
- Rotation on conics
A conic involution resembling the usual rotation, but defined on a conic. And a recipe to create many "Poncelet" polygons tangent to two conics.
- Rotations On Quadrangle Vertices
Calculating the composition of rotations on the vertices of a quadrangle.
- Rotations On Quadrangle Vertices circum
Calculating the composition of rotations on the vertices of a circumscribable, on a circle, quadrangle.
- Roulette (inner)
A special case of a roulette resulting by a point rigidly attached to a disc which rolls inside a circle.
- Roulette (outer)
A special case of a roulette resulting by a point rigidly attached to a disc which rolls outside a circle.
- S-Triangles created : 05/05/2008 euc
Triangles related by their deltoids.
- Salinon
The geometry of a shield of ancient Greeks.
- Seesaw
The construction of a seesaw on a circular arc.
- Geometric Series
The convergence of the geometric series for q = 1/3, illustrated geometrically.
- Sequence of triangles
A remarkable sequence of triangles created for each triangle and each triangle center of it.
- Sewing
The feed mechanism of a sewing machine, realized through motor-objects of EucliDraw.
- Sewing (2)
Slider-Crank thread and needle guiding mechanism of a sewing machine.
- Sierpinsky Fractal
Application of the tool [Fractal-Socket] of EucliDraw to produce a well known fractal.
- Similarity created : 22/06/2006, updated : 31/08/2009, 02/02/2011 euc
A first discussion of the similarity transformation together with an interesting application.
- Similarly gliding created : 22/06/2006, updated : 02/02/2011
Triangles varying with all their vertices on fixed lines while remaining similar all the time.
- Similarly rotating created : 22/06/2006, updated : 02/02/2011
Triangles with a fixed vertex and varying the other vertices on lines while remaining similar all the time.
- Similar Inscribed triangles
Introduction to similarities through two similar triangles with a common vertex.
- Similar Inscribed created : 22/06/2006, updated : 02/02/2011
Introduction to similarities through two similar quadrangles with a common vertex.
- Homotheties and Similarities [New pdf-version] 22/06/2006, 01/12/2018, 30/05/2021
Contents:
1 Homotheties ,
2 Homotheties and triangles ,
3 Homotheties with different centers ,
4 Homotheties and translations ,
5 Representation and group properties of homotheties ,
6 Similarities, general definitions,
7 Similarities defined by two segments ,
8 Similarities and orientation ,
9 Similarities and triangles ,
10 Triangles varying by similarity ,
11 Relative position in similar figures ,
12 Polygons on the sides of a triangle ,
13 Representation and group properties of similarities ,
14 Logarithmic spiral and pursuit curves.
Related topics :
Isometries or Congruences,
Menelaus' theorem,
- Similarity Centers
Properties of the two similarity centers of two circles exterior to each other.
- Simple Geometric Locus
Simple Geometric Locus, generated by the appropriate tool of EucliDraw.
- Simple Geometric Locus (2)
Simple Geometric Locus, generalizing the previous one.
- Simson line of a triangle created : 22/06/2006, updated : 05/05/2008, 10/10/2008 euc
Projecting a point on the circumcircle onto the sides of a triangle.
- Simson 3 Lines
A configuration involving the Simson lines and the Morley triangle.
- Simson lines at right angles created : 27/10/2007, updated : 31/01/2008 euc
Diametral points on the circumcircle have corresponding Simson lines orthogonal, intersecting on the Euler circle.
- Generalized Simson Lines created : 22/06/2006, updated : 05/05/2008, 10/10/2008 euc
Instead of orthogonal projections on triangle sides producing the well known Simson lines, here another recipe.
- A parallel to the Simson line created : 22/06/2006, updated : 05/05/2008, 10/10/2008 euc
A line parallel to the Simson line.
- The angle between two Simson lines
A property of the angle of two Simson Lines relative to the same triangle.
- Simson Related
An exercise on Simson lines involving the antiparallel triangle.
- Three Simson lines of a triangle
The triangle formed by the Simson lines of three points on the circumcircle of a triangle.
- Generalized Simson Lines Envelope
The envelope curve of all generalized Simson lines for a fixed angle.
- Simson Variant Locus
Rotate a triangle in its circumcircle and produce a locus with kinetic conditions.
- Soliton
An example of a traveling soliton.
- Solitons interfering
The phenomenon of interference of two traveling solitons.
- Somov
Somov's slider-crank mechanism for tracing Cassinian ovals.
- Spiral Tiling
Algorithm to place M parallelograms in a spiral-like way around the first which serves as prototype for the tiling.
- Spiral Tiling Hexagon
Algorithm to place M regular-hexagonal tiles in a spiral-like way around the first tile.
- A property of the square
A simple property of secants of a square.
- Extract the square root
The classic easy way to extrac geometrically the square root of a number.
- Square root mechanism
A linkage used to extract mechanically the square root of a number.
- Squares through
Given four points A,B,C,D, construct a square, whose sides pass through these points.
- Squares through 4_points
The 6 squares with sides (or their prolongations) passing through four given poins A, B, C, D. The construction is done using the tool of scheme-sockets.
- Square construction
Find the square having two points on opposite sides and two points on the diagonals.
- Squares-Dissection 11/08/2009, 02/02/10 euc
Dissect two squares in pieces and rearrange them into one square.
- Square in Square
Square into another square defines four areas, the two opposite having area sum equal to the other two.
- Squares Combination
The locus of points whose sum of weighted distances from some points remains constant.
- Staircase automatic
Automatic Staircase. Shown the principle which uses the most elementary curve of constant width.
- Steiner Chain
Steiner's theorem on properties of chains of circles, based on inversions.
- Steiner Chain
As before, just another viewpoint.
- Steiner Chain
As before, just a third viewpoint.
- Steiner Ellipses created : 22/06/2006, updated : 31/08/2009 euc
Steiner Ellipses of a triangle and how to construct them quickly with EucliDraw.
- A theorem of Steiner
A theorem of Steiner on the generation of ellipses.
- A theorem of Steiner (2)
A theorem of Steiner relating to Pascal's theorem on inscribed hexagons.
- Steiner lines
Lines parallel to Wallace-Simson lines, passing through the orthocenter of a triangle.
- Steiner point created : 08/08/2007, updated : 31/01/2008 euc
The Steiner point, fourth intersection point of the circumcircle with the outer Steiner ellipse. An easy construction.
- Steiner reflected
Creating the point corresponding to a given Steiner line by reflecting this line on the sides of the triangle.
- Stereographic projection
The classical map serving as prototype for "good parametrizations" of conics.
- Stewart's theorem
Stewart's theorem on the length of Cevians of a triangle.
- Strophoid
A theorem on strophoids taken from an old french book (Aubert-Papelier).
- Subnormal of a conic created : 02/01/12
The length of the projection of the normal of a conic on the axis carrying the focal points.
- Symmedian [New pdf-version] 29/12/2007, 01/12/2018, 15/05/2021
Contents:
1 Symmedian and symmedian point ,
2 Antiparallels ,
3 Harmonic quadrangle ,
4 Second Brocard triangle ,
5 Gergonne point of the triangle ,
6 First Lemoine circle ,
7 Adams’ circle ,
8 Conics through parallels ,
9 Characterization of the first Lemoine circle ,
10 Second Lemoine circle ,
11 Inscribed rectangles ,
12 Medial and Orthic triangle intersections ,
13 Vecten squares of the triangle ,
14 Arzt parabolas.
Related topics :
Apollonian circles,
Artzt parabolas,
Desargues' theorem,
Trilinear coordinates,
Tucker circles.
- Symmedian property
A property characterizing the symmedian lines of a triangle.
- Symmedian - Vecten
Symmedians and their relation to the Vecten configuration of the triangle.
- Symmetric hexagons
Circum/Inscribed conics of symmetric hexagons.
- Symmetric hexagons All
How to classify all symmetric hexagons.
- Symmetric Hexagons Special
Study of a special class of symmetric hexagons.
- Symmetric Hexagons Nested
Study of the special case of symmetric nested (or derived) hexagons through their side-middles.
- Symmetric Octagons Nested
Picture of the special case of symmetric nested (or derived) octagons through their side-middles.
- Symmetric triangles with respect to a point
Pairs of triangles, symmetric with respect to a point and a related conic.
- Symmetries On Vertices
Study of the composition of symmetries on the vertices of a rectangle.
- Symmetries On Vertices Even
Study of the composition of symmetries on the vertices of an even sided polygon.
- Symmetries On Vertices Even (2)
Study of the composition of symmetries on the vertices of an even sided polygon. Symmetric polygons.
- Symmetries On Vertices Odd
Study of the composition of symmetries on the vertices of an odd sided polygon.
- Tangent Cuts Circle
Envelope of tangent-cuts from a circle to two other circles, the three circles building a circle-pencil.
- Tangent Cuts Envelope
Envelope of tangent-cuts from a conic to another conic.
- Tangents envelope
Envelope of contact-joins of tangents from a circle to two other circles, the three circles building a circle-pencil.
- Tangent member
Find the members of a circle pencil that are tangent to a given circle.
- All conics tangent to four given lines
Construction of all conics tangent to four given lines.
- Circumcircle of the tangential triangle
The circumcircle of the tangential triangle, its location on the Euler line and location of X(25).
- Tangram
Tangram realization through EucliDraw.
- Terquem General
Generalization of the Terquem transformation using two special points and a conic.
- Tetradivision
Construct rhombi inside triangles.
- Thales theorem
A first discussion of the classical theorem and its applications.
- Thales theorem II created : 22/06/2006, updated : 01/05/2007, 10/10/2008, 31/08/2009 euc
A further discussion of the classical theorem and its applications.
- Thales application
An application of Thales theorem.
- A generalization of Thales Theorem created : 22/06/2006, updated : 31/08/2009 euc
Generalization of Thales theorem on the segments cut of on a pencil of lines by two or more parallels.
- Thales Parabola created : 22/06/2006, updated : 31/08/2009 euc
A parabola related to two lines and a point not belonging to these lines.
- Thales remarks created : 22/06/2006, updated : 31/08/2009, 02/02/2011 euc
Remarks on Thales theorem escalating from the very simple to the more advanced.
- Thebault
Constructing squares on the sides of a parallelogram. Thebault's theorem.
- Three angles created : 11/08/2009 euc
A property of three angles on a common chord.
- Three bar dwell
Construction of a dwell mechanism.
- Three Collinear Points
An interesting exercise related to a circle.
- Three Collinear Points (2)
As before but more general, involving conics.
- Three circles problem
A problem involving three circles and leading to conic-construction.
- Three circles problem (2)
A conic construction involving three circles and a radical axis.
- Three circles problem (3)
Various curves related to the previous construction, involving three circles.
- Three diameters created : 22/06/2006, updated : 01/05/2007, 10/10/2008 euc
Application of the property of Wallace-Steiner lines.
- Three lines 12/12/2009, 02/02/10 euc
Basic aspects of the configuration consisting of three lines passing through the same point.
- Tractrix
A famous curve, its equation and geometric generation.
- Tractrix Mamikon
Application of the Mamikon device to calculate the area under a tractrix.
- Trapezium 12/12/2009, 02/02/10 euc
Trapezium's basic properties .
- Trapezium II 11/08/2009, 02/02/10 euc
Trapezium's some more properties.
- Trapezium (2) 11/08/2009, 02/02/10 euc
A property of trapezia.
- Trapezium Division
Trapezium's division in two equal (in area) parts, by a line passing through the middle of a non-parallel side.
- Producing triangles with the same centroid
Extending the sides of a triangle proportionally and building triangles with the same centroid.
- Triangle Bisectors
A study of some lines associated with the bisectors of the angles of a triangle.
- Two triangle constructions involving bisectors created : 22/06/2006, updated : 12/12/09 euc
Two triangle constructions involving bisectors and a reference to an interesting account of triangle constructions.
- Conics circumscribing a triangle-I
A study of the conics circumscribing a triangle.
- Conics circumscribing a triangle-II
A study of the conics circumscribing a triangle. Just another aspect.
- Triangles Circumscribing Parabolas
The basic properties of triangles having their sides tangent to a parabola.
- Triangle Conics created : 08/08/2007, updated : 31/01/2008, 05/05/2008 euc
A discussion of the pairing of conics inscribed/circumscribed on a triangle.
- Triangle conics intersections
The fourth intersection point of two triangle conics. How to find it easily.
- Triangle Construction from A, a, r_{b}
To construct a triangle from A, a and r_{b}.
- Triangles with given pivot created : 12/12/2009 euc
Constructing a triangle inscribed in a conic and having a given point as pivot for this conic.
- Triangles with given pivot II created : 12/12/2009 euc
Constructing a special triangle inscribed in a conic and having a given point as pivot for this conic.
- Triangle Glide Reflexions
Composition of three reflexions in general position.
- Projectivities related to a triangle
Some basic relations between in/circum-conics of a triangle, cevians and trilinear polars expressed through projectivities.
- Triangle Rotations Product
Composition of three rotations about the vertices of a triangle.
- Line of triangles
Varying a triangle so that its vertices glide on three lines.
- Triangle Symmetrization
Transforming an arbitrary triangle to an equilateral in its circumcircle.
- Triangles Line
A kind of line whose points are triangles.
- Trilinear Polar created : 22/06/2006, updated : 05/05/2008, 01/09/2009 euc
The line of perspectivity of the Cevian triangle of a point with respect to a triangle.
- Find triangle with given trilinear polar created : 31/01/2008, updated : 01/09/2009 euc
Given a conic and a line to inscribe a triangle in the conic having the given line as trilinear polar.
- Conics generating parallel trilinear polars created : 22/06/2006, updated : 05/05/2008 euc
Triangle conics whose points have trilinear polars parallel to a fixed direction.
- Trilinear Polar Envelope
A problem of finding the envelope of trilinear polars.
- Trilinear coordinates created : 22/06/2006, updated : 05/05/2008 euc
Definition and first properties of the standard trilinear coordinates relative to a triangle.
- Trilinears and others created : 22/06/2006, updated : 12/12/09 euc
Relations of trilinear coordinates to some other systems of coordinates associated to a fixed triangle.
- Tripole
The point M with respect to which a line is the trilinear polar for a given triangle.
- Trireme
Picture of a greek trireme, animated through a motor-object of EucliDraw.
- Tritangent circles [New pdf-version] 08/09/2007, 12/07/2019, 28-11-2021
Contents:
1 Inscribed, Escribed, Incenter and Excenters ,
2 Tangents from the vertices ,
3 Relations between the radii 5
4 Radii of the tritangent circles related to side-lengths ,
5 Relations between angles and side-parts ,
6 Heron’s triangle area formula ,
7 A symmetric equilateral hexagon ,
8 Euler’s theorem for the incenter/excenter ,
9 Some properties of the bisectors ,
10 Euler’s line and circle of the triangle ,
11 Some properties of Euler’s circle ,
12 Feuerbach’s theorem.
- Van Aubel
Van Aubel' s theorem on squares on the sides of a quadrangle.
- Van Aubel (2)
Van Aubel' s theorem on squares additional properties and proofs.
- Vecten
The Vecten configuration of a triangle.
- Vecten (2)
More on the Vecten configuration of a triangle.
- Vecten (3)
More on the Vecten configuration of a triangle.
- Vecten (4)
More on the Vecten configuration of a triangle.
- Vecten (5)
More on the Vecten configuration of a triangle.
- Vecten Inequality
A natural inequality connected with the Vecten diagram of a triangle.
- Vecten diagram of an equilateral triangle
A simple case of measurement of the perimeter-ratio of two equilateral triangles.
- Vecten tiles
Glueing Vecten tiles together makes a grid.
- Wallace-Simson line
Some remarks concerning the Wallace-Simson lines of a triangle.
- Weighted Sums of Distances
On the locus of points whose sums of weighted distances from some fixed points remains constant.
- Weird Sinus Zoom
A picture on the use of the Zoom-inside-a-rectangle-tool of EucliDraw.
- Woo's Construction
A simply construction of the incircle of an arbelos.
- Yagci's problem
Solution of a problem related to the Brocard points of a triangle.
- Yates Trisector
A linkage to trisect angles, originated by Laisant and improved by Yates.