Question 56
A triangle $ABC$ is given. A circle $\Gamma$ passes through vertex $A$ and is tangent to side $BC$ at point $P$. The circle $\Gamma$ intersects sides $AB$ and $AC$ at points $M$ and $N$, respectively. Prove that (minor) arcs $\widehat{MP}$ and $\widehat{NP}$ are equal if and only if $\Gamma$ is tangent to the circumcircle of $\triangle ABC$ at $A$.