# Geometry (Thailand Math POSN 3rd round)

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.

Tucker Circle: see Tucker Circle

Taylor Circle: see Taylor Circle

Write a full solution.

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

2. 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.

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

4. 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 Note by Samuraiwarm Tsunayoshi
6 years, 3 months ago

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