Question 36
Let $ABC$ be a triangle with $\angle BAC = 90^\circ$ with the altitude $AH$ ($H \in BC$). A circle $(\omega)$ passes through $B, C$ and cuts the segments $AB, AC$ at $M, N$ respectively. Circle $(\omega)$ also cuts the line $AH$ at $D, E$ ($D$ lies between $A, H$). Suppose that $DE = AH\sqrt{5}$, prove that the circumcircle of triangle $HMN$ is tangent to $BC$.