The line at infinity in turn is the trilinear polar of the center O of the equilateral. Thus, we have a procedure determining the incircle by taking repeatedly trilinear polars of points:

- Start with the system (A

- Take the trilinear polar L

- For every point P

In fact, define F by the requirements to map the vertices of the equilateral {A

Since projectivities respect cross ratios of points on a line, the line at infinity L

This property of inconics can be thus formulated:

The proof follows from the very definition of the Gergonne point K as intersection point of the lines joining the contact points with opposite vertices. Thus apply the previous general results to this special case.

Note that the incircle is the circumcircle of triangle A'B'C' of contact points with the sides of ABC. The Gergonne point with respect to ABC coincides with the symmedian point K with respect to A'B'C'. On the other side the trilinear polar of K with respect to triangles ABC and A'B'C' is the same line (see TrilinearPolar.html ). Since the trilinear polar of the symmedian point with respect to the circumcircle is the

See CircumconicsTangents.html for a discussion on an analogous generation of the circumconics of a triangle.

CircumconicsTangents.html

HyperbolaPropertyParallels.html

HyperbolaPropertyParallels2.html

IncircleTangents.html

IsogonalGeneralized.html

TrilinearPolar.html

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