Problem

Source: Chinese TST 2007 5th quiz P1

Tags: geometry, circumcircle, parallelogram, perpendicular bisector, geometry proposed



Let convex quadrilateral ABCD be inscribed in a circle centers at O. The opposite sides BA,CD meet at H, the diagonals AC,BD meet at G. Let O1,O2 be the circumcenters of triangles AGD,BGC. O1O2 intersects OG at N. The line HG cuts the circumcircles of triangles AGD,BGC at P,Q, respectively. Denote by M the midpoint of PQ. Prove that NO=NM.