Gergonne Triangle and Point
The Gergonne triangle of ABC is denoted by the vertices TA, TB and TC that are the three points where the incircle touches the reference triangle ABC and where TA is opposite of A, etc. This triangle TATBTC is also known as the contact triangle or intouch triangle of ABC. The incircle of ABC is the circumcircle of TATBTC. The three lines ATA, BTB and CTC 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.
Read more about this topic: Incircle And Excircles Of A Triangle
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