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.

Adam Circle: see Adams' Circle

Tucker Circle: see Tucker Circle

Taylor Circle: see Taylor Circle

Write a full solution.

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

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

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

  4. Let TT be a center of Taylor circle in ABC\triangle ABC. Prove that AT2ha2=BT2hb2=CT2hc2AT^{2} - h_{a}^{2} = BT^{2}-h_{b}^{2} = CT^{2}-h_{c}^{2} where ha,hb,hch_{a},h_{b},h_{c} are altitudes of ABC\triangle ABC from vertices A,B,CA,B,C respectively.

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

Note by Samuraiwarm Tsunayoshi
4 years, 6 months ago

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