Problem 561: Triangle, Cevian, Incenter, Perpendicular, 90 Degrees.
Level: High School, College, Math Education

The figure shows a triangle ABC with a cevian BD. Points E, F, and O are the incenters of triangles ABD, BCD, and ABC, respectively. If OG is perpendicular to AC, prove that the measure of angle EGF is 90 degrees.