ArtztSteiner

1. Artzt parabola and Steiner outer ellipse

Artzt parabolas of a triangle (of first kind) are related to the outer Steiner ellipse. Next figure illustrates the connexion.
Start with the centroid G of ABC, the median AG, G' the middle of AG and Q' the other intersection point of the median with the outer Steiner ellipse.
Take a point P on the basis BC of triangle ABC and draw a parallel PP' to the median AG.
Define P' to be the intersection point of this parallel with the line parallel to the base BC through G'.
Finally define Q to be the other intersection point of line Q'P with the outer Steiner ellipse.
[1] Triangles AP'G and GPQ' are equal. GP'QP is a trapezium.
[2] The intersection point W of the diagonals of this trapezium describes the Artzt parabola p_BC of triangle ABC (see Artzt.html ).

A very simple proof of these facts results by applying an affinity to a trivial case of the theorem concerning an equilateral triangle. In ArtztIsosceles.html there is an analogous discussion for the case of an isosceles triangle. The results there apply in particular to the case of an equilateral triangle. Then using the affinity which maps the vertices + centroid of the equilateral to corresponding vertices + centroid of the general triangle we obtain the proofs of the properties stated.

2. Homography mapping ellipse to parabola

As is well known (see Projectivity.html ) a projectivity is defined by prescribing the image points {Y_i, i=1,..4} of four points {X_i, i=1,...,4} in general position. Thus there is a unique projectivity F with the properties:
(i) F fixes points {B,C, G} and
(ii) F maps A to W₀.
- It follows easily (exercise) that F fixes every point of line BC and also leaves invariant every line passing through G (maps such a line into itself fixing G and its intersection point with BC).
- Using this and the invariance of the medial line AG one sees easily that every line parallel to BC maps to a line also parallel to BC. In particular the parallel to BC through A maps to the parallel to BC through W₀.
- Using cross ratios allong line AG one sees easily that the tangent line t of the ellipse at Q' maps via F to the line at infinity.
- From the previous result follows also immediately that the tangents {t_B, t_C} to the ellipse at {B,C}, map via F to sides {BA, CA} respectively.
- From these facts follows, that points {A,B,C} and the tangents to the ellipse at these points map to {W₀,B,C} and the tangents to the parabola at these points correspondingly. Thus, also the ellipse maps via F to the Artzt parabola c.

3. Generalization to arbitrary triangle conic

The outer Steiner ellipse is the distinguished model of the more general triangle conics (see TriangleConics.html ) circumscribing a given triangle and being determined through a particular point not on the side-lines of the triangle, called perspector of the triangle conic. The various triangle conics are permuted by projectivities fixing the triangle vertices and mapping their perspectors, one to the other. Thus, all projective properties obtained for a particular triangle conic (as f.e. the outer Steiner ellipse) can be transferred by appropriate projectivities to any other triangle conic. The file ArtztSteiner2.html examines how the properties studied here transfer to a general triangle conic determined by an arbitrary perspector K.

1. Artzt parabola and Steiner outer ellipse

2. Homography mapping ellipse to parabola

3. Generalization to arbitrary triangle conic

See Also