Problem 681: Gergonne Point, Triangle, Incircle, Tangency points, Concurrent lines, Semi-perimeter, Ceva theorem. Level: High School, SAT, College Geometry

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