Theorems allowed to use: Basic theorems, Isogonal conjugate, Symmedian point, Gergonne point, Nagel point, Adam Circle, Tucker Circle, Tucker Hexagon, Taylor Circle.

Gergonne point: 3 lines that pass through the vertex and the contact point of incircle are concurrent at Gergonne point.

Nagel point: 3 lines that pass through the vertex and the contact point of excircle are concurrent at Nagel point.

Adam Circle: see Adams' Circle

Tucker Circle: see Tucker Circle

Taylor Circle: see Taylor Circle

Write a full solution.

Prove that Gergonne point of \(\triangle ABC\) is a symmedian (Lemoine point) of Gergonne triangle.

Prove that 3 lines that pass through excenters of \(\triangle ABC\) and midpoints of sides of \(\triangle ABC\) that are closest to excenters are concurrent.

Let \(\triangle DEF\) and point \(G\) be Gergonne triangle and Gergonne point of \(\triangle ABC\), and \(D,E,F\) are opposite to \(A,B,C\).

Let \(T\) be a center of Taylor circle in \(\triangle ABC\). Prove that \[AT^{2} - h_{a}^{2} = BT^{2}-h_{b}^{2} = CT^{2}-h_{c}^{2}\] where \(h_{a},h_{b},h_{c}\) are altitudes of \(\triangle ABC\) from vertices \(A,B,C\) respectively.

This note is part of Thailand Math POSN 3rd round 2015

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