Let $ABC$ be an acute triangle with $AB<AC<BC, c$ it's circumscribed circle and $D,E$ be the midpoints of $AB,AC$ respectively. With diameters the sides $AB,AC$, we draw semicircles, outer of the triangle, which are intersected by line $D$ at points $M$ and $N$ respectively. Lines $MB$ and $NC$ intersect the circumscribed circle at points $T,S$ respectively. Lines $MB$ and $NC$ intersect at point $H$. Prove that: a) point $H$ lies on the circumcircle of triangle $AMN$ b) lines $AH$ and $TS$ are perpedicular and their intersection, let it be $Z$, is the circimcenter of triangle $AMN$