Square inscribed in a right triangle problem

Let A be a point on a fixed semicircle with diameter BC. MNPQ is a square such that $M \in AB, N \in AC, P \in BC, Q \in BC$. Let D be the intersection of BN and CM and E be the center of the square. Prove that as A varies, DE always passes through a fixed point.

The fixed point is the midpoint of the semicircle. Any suggestion ?

Edit : I have proved that AE is the angle bisector of angle BAC and AD passes through the midpoint of BC.

I think that DE also passes through the feet of the altitude to BC. May be harmonic bundle is useful. (D,H,E,J) = -1 ?

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