Concyclic Points Related to a Midpoint and the Incircle
Let ABC be a triangle and M be the midpoint of AB. Let A', B', and C' be the intersections of the incircle with BC, AC, and AB, respectively. Let γ be a circle that passes through A and B and contains A' and B' in its interior. Let A'B' intersect γ at P and Q. Then P, Q, C', and M are concyclic.