Gergone

Gergone's point

The universal property of Gergone's point G of a triangle ABC.
This is the point which is common to all joins AA', BB', CC', of the vertices with the opposite contact point with the incircle. The universal property is the following. Consider the homography F, fixing the vertices [F(A)=A, F(B)=B, F(C)=C'] and mapping G to another point G'. Then take the image c' = F(c) of the incircle under this homography. c' is a conic tangent to the triangle and G' is the intersection point of the joins AA'', BB'', CC'' of the vertices to the opposite contact points. It is AA'' = F(AA') etc. Every conic tangent to the triangle comes from such an homography.

Switch to [selection-tool] (CTRL+1). Catch and modify A, B, C, G'.

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