We consider the equilateral triangle T defined on E by the unit vectors e

Notice that x must not lie on the sides (prolongations) of the equilateral. This is the necessary condition for the definition of the projectivity F

The previous remark reveals also the various parts of the plane where the corresponding c

The above argument can be transferred, through conjugation via an affinity, to an arbitrary triangle T=ABC. In fact, consider the affinity F mapping the vertices of a fixed equilateral to the vertices of ABC and the center of the equilateral to the centroid G of ABC. Then the in/circum-circle of the equilateral are mapped by F to the corresponding inner/outer Steiner ellipses of ABC (see Steiner_Ellipse.html ). Then use projectivities of the form:

G

Note that each of these circumconics can be directly defined through the projectivity mapping the vertices of the equilateral to the vertices of the triangle and the center of the equilateral to point y. to define all circumconics of ABC.

The above discussion shows that for all y inside the inner Steiner ellipse we get elliptic circumconics. For y on the inner Steiner ellipse we get parabolas circumscribed and for y outside the inner Steiner conic we get hyperbolas (see TriangleCircumconics.html ).

CrossRatio0.html

CrossRatioLines.html

GoodParametrization.html

Harmonic.html

Harmonic_Bundle.html

HomographicRelation.html

HomographicRelationExample.html

Pascal.html

PascalOnQuadrangles.html

PascalOnTriangles.html

Steiner_Ellipse.html

TriangleCircumconics.html

TriangleCircumconics2.html

TriangleProjectivitiesPlay.html

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