A Concurrency from the Midpoints of Line Segments through the Circumcenter

Let ABC be a triangle with circumcenter O. Let A', B', and C' be the perpendicular projections of A, B, and C onto BC, CA, and AB, respectively. Let , , and . Let A'', B'', and C'' be the midpoints of AD, BE, and CF, respectively. Then A'A'', B'B'', and C'C'' are concurrent.