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

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.

(i) F fixes points {B,C, G} and

(ii) F maps A to W

- 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 W

- 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

- From these facts follows, that points {A,B,C} and the tangents to the ellipse at these points map to {W

Artzt.html

Artzt_2.html

Artzt2.html

ArtztCanonical.html

ArtztIsosceles.html

ArtztSteiner2.html

Artzt_Generation.html

Artzt_Generation2.html

HyperbolaPropertyParallels.html

ParabolaFromProjections.html

ProjectivityFixingVertices.html

Steiner_Ellipse.html

TriangleCircumconics.html

Trilinear_Polar.html

ParabolaChords.html

ParabolasFromEqualSegs.html

ParabolaSkew.html

ReflexionsOfLine.html

ThalesRemarks.html

IsoscelesIntersection.html

Symmedian_1.html

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