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 AT2−ha2=BT2−hb2=CT2−hc2AT^{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

#Geometry

Note by Samuraiwarm Tsunayoshi
6 years, 1 month ago

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