Problem 682: Gergonne External Point, Triangle, Excircle, Tangency points, Concurrent lines, Semiperimeter, Ceva theorem. Level: High School, SAT, College Geometry

The figure shows a triangle ABC with the excircle E_A tangent to the sides at the points A', B' and C'. Prove that AA', BB', and CC' are concurrent at a point G_A called Gergonne external point.
Joseph Diez Gergonne (1771-1859) was a French mathematician and geometer who founded Annales de mathematiques in 1810.