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# 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
2 years, 5 months ago