Online Geometry Problem 792: Right Triangle, Altitude, Angle Bisector, Three Incircles, Incenters, Five Concyclic Points, Circles. Level: High School, SAT Prep, College, Mathematics Education

The figure below shows a right triangle ABC with the incenter I and the altitude BH. BD and BE are bisectors of angles HBC and ABH, respectively. I₁ is the incenter of the triangle AHB and I₂ is the incenter of triangle BHC. Prove that the points E, I₁, I, I₂, and D are concyclic.