Incircle and Excircles of A Triangle - Gergonne Triangle and Point

Gergonne Triangle and Point

The Gergonne triangle of ABC is denoted by the vertices T_A, T_B and T_C that are the three points where the incircle touches the reference triangle ABC and where T_A is opposite of A, etc. This triangle T_AT_BT_C is also known as the contact triangle or intouch triangle of ABC. The incircle of ABC is the circumcircle of T_AT_BT_C. The three lines AT_A, BT_B and CT_C intersect in a single point, the triangle's Gergonne point Ge - X(7). Interestingly, the Gergonne point of a triangle is the symmedian point of its Gergonne triangle. For a full set of properties of the Gergonne point see.

The touchpoints of the three excircles with segments BC,CA and AB are the vertices of the extouch triangle. The points of intersection of the interior angle bisectors of ABC with the segments BC,CA,AB are the vertices of the incentral triangle.