Problem 1104: Right Triangle, Incircle, Circumcircle, Inscribed Circle, Radius, Tangent. Level: High School, SAT Prep, College Geometry

In a right triangle ABC (see the figure below) with circumcircle Q, the incircle O (radius r) is tangent to AB, BC, and AC at D, E, and F, respectively. Circle O₁ (radius x) is tangent to AB, BC and arc AC at H, M, N, respectively. Prove that x = 2r = a + c - b.