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 is a symmedian (Lemoine point) of Gergonne triangle.
Prove that 3 lines that pass through excenters of and midpoints of sides of that are closest to excenters are concurrent.
Let and point be Gergonne triangle and Gergonne point of , and are opposite to .
Let be a center of Taylor circle in . Prove that where are altitudes of from vertices respectively.
This note is part of Thailand Math POSN 3rd round 2015