SHARYGIN 2006 Problem 15

A circle is circumscribed around triangle $ABC$ and a circle is inscribed in it, which touches the sides of the triangle $BC,CA,AB$ at points $A_1,B_1,C_1$ , respectively. The line $B_1C_1$ intersects the line $BC$ at the point $P$ , and $M$ is the midpoint of the segment $PA_1$ . Prove that the segments of the tangents drawn from the point $M$ to the inscribed and circumscribed circle are equal.