Geometry Problem 1453: Two Semicircles, Cyclic Quadrilateral, Concyclic Points

Semicircles of diameters AB and CD intersect at E (see the figure below). FG is a line through E. BF and CG meet at H, BF and CE meet at M, and CG and BE meet at N. Prove that ENHM is a cyclic quadrilateral.
Geometry Problem 1453: Two Semicircles, Cyclic Quadrilateral, Concyclic Points

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Geometry Problem 1453 Solution(s)