Problem 35. Incenters and Inradii in Cyclic Quadrilateral. Level: High School, SAT Prep, College

The figure shows a cyclic quadrilateral ABCD. E, F, G, and H are the incenters and r_a, r_b, r_c, and r_d are the inradii of the triangles ABD, ABC, BCD and ACD respectively. Prove that EFGH is a rectangle and r_a + r_c = r_b + r_d.
Solution /vcomment.