Question 133
Let $AB\Gamma$ be a triangle with circumcircle $\omega(O, R)$. A circle $\gamma$ passes through $O$ and $B$ and is tangent to the line $AB$ at $B$. Let the circle $\gamma$ meet the circle $\omega(O, R)$ for a second time at $P \neq B$. A circle passes through $P$ and $\Gamma$ and is tangent to the line $A\Gamma$ at $\Gamma$ and meets the circle $\gamma$ at $M \neq P$. Prove that: $MP = M\Gamma$.