Question 127
The point $M$ is the midpoint of the side $BC$ of the triangle $ABC$. A circle passes through $B$, is tangent to the line $AM$ at $M$ and intersects the segment $AB$ for the second time at the point $P$.
Prove that the circle passing through $A$, $P$ and the midpoint of the segment $AM$ is tangent to the line $AC$.
Prove that the circle passing through $A$, $P$ and the midpoint of the segment $AM$ is tangent to the line $AC$.