All these properties follow trivially by considering the projectivity F mapping the vertices of an equilateral A

F maps the circumcircle of the equilateral to the circumcircle of the triangle ABC.

F mas also the incircle of the equilateral to the Brocard ellipse of ABC.

The antipodal of A

Notice the concurrence of the four lines: tangents at opposite vertices and opposite sides (extended) at the same points of the Lemoine axis. This follows from the corresponding properties of the pre-images of these lines under projectivity F and the fact that F maps the line at infinity to the Lemoine axis of the triangle.

This is a preliminary review of a rich in content subject to be discussed in more detail at a later moment.

CrossRatio.html

CrossRatio0.html

CrossRatioLines.html

GoodParametrization.html

Harmonic.html

Harmonic_Bundle.html

HomographicRelation.html

HomographicRelationExample.html

IncircleConjugate.html

Pascal.html

PascalOnTriangles.html

ProjectivityFixingVertices.html

Steiner_Ellipse.html

TriangleCircumconics.html

TriangleCircumconics2.html

TriangleProjectivitiesPlay.html

TrilinearPolar.html

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