Concyclic Points Derived from Midpoints of Altitudes
Let ABC be a triangle and D, E, F be the feet of the altitudes. Let D', E', F' be the midpoints of the segments AD, BE, CF. Let the line E' F' meet the lines AB and CA at the points B' and C'', let the line F'D' meet the lines BC and AB at the points C' and A'', and let the line D'E' meet the lines CA and BC at the points A' and B''. If X, Y, Z are the circumcenters of triangles AB'C'', BC'A'', and CA'B'', respectively, then the points D', E', F', X, Y, Z are concyclic.