Sharygin 2010, correspondence round, problem 12
Sharygin 2010, correspondence round, problem 12 is a plane geometry problem from the Sharygin Geometry Olympiad 2010, correspondence round, problem 12 (grade 9): the altitude of a right triangle and a tangent.
Problem
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 $B_0$ be the reflection of $B$ with respect to the point $H$. The perpendicular to $AB$ erected at $B_0$ meets the circle in a point $K$. Prove that: a) $B_0M \parallel BC$; b) $AK$ is tangent to the circle.
Source Sharygin Geometry Olympiad 2010, correspondence round, problem 12 (grade 9)
The diagram
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