SHARYGIN 2010 Problem 12

Let $AC$ be the greatest leg of a right triangle $ABC$, and $CH$ be the altitude to its hypotenuse. The circle of radius $CH$ centered at $H$ intersects $AC$ in point $M$. Let a point $B'$ be the reflection of $B$ with respect to the point $H$. The perpendicular to $AB$ erected at $B'$ meets the circle in a point $K$. Prove that: a) $B'M \parallel BC$; b) $AK$ is tangent to the circle.