Problem 582: Quadrilateral, Diagonals, Incenters, Metric Relations. Level: Mathematics Education, High School, SAT Prep, College

The figure shows a quadrilateral ABCD.
Diagonals AC and BD meet at O. If E, F, G, and H are the incenters of
triangles AOB, BOC, COD, and AOD, respectively, prove that EF^{2} +
GH^{2} = EH^{2} + FG^{2}.