Notice that the circumconics of the equilateral can be obtained through the various projectivities (denoted here by

Since F is not fixing the line at infinity it is not an affinity. In particular ratios of areas are not preserved and the outer Steiner ellipse of triangle ABC, which is its circumcircle is not mapped to the outer Steiner ellipse of A'B'C'. To obtain this conic one has to take E such that the corresponding trilinear polar L(E) maps under F (i.e. line L(E')) to the line at infinity.

The file ProjectivityFixingVertices.html contains a study of the projectivity D-to-E for the case of the equilateral.

ProjectivityFixingVertices.html

Steiner_Ellipse.html

TriangleCircumconics.html

TriangleConics.html

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