Online Geometry Problem 794: Right Triangle, Altitude, Three Incenters, Circumcircles, Incircle, Congruence. Level: High School, SAT Prep, College, Mathematics Education

The figure below shows a right triangle ABC with the incenter I and the altitude BH. D and E are the incenters of triangles AHB and BHC, respectively. Prove that the incircle of triangle ABC and the circumcircles of triangles BDE and DEI are congruent.