A Concurrency Generated by Lines through the Orthocenter and Circles about a Triangle's Sides

Let ABC be a triangle, P be an interior point, and H be the orthocenter. Let A', B', and C' be the internal intersections of AP, BP, and CP with the circles whose diameters are BC, AC, and AB, respectively. Let A'H, B'H, and C'H intersect the circles whose diameters are BC, AC, and AB again at A'', B'', and C''. Then AA'', BB'', and CC'' are concurrent.