Question 31
Triangle $ABC$ is inscribed in circle $\omega$. Line $\ell$ is tangent to $\omega$ at $A$. Points $B_1$ and $C_1$ lie on $\ell$ such that rays $CA$ and $BA$ bisect $\widehat{BCB_1}$ and $\widehat{CBC_1}$, respectively. Segments $BB_1$ and $CC_1$ intersect at $P$. The line through $P$ parallel to segment $BC$ intersects sides $AC$ and $AB$ at $B_2$ and $C_2$, respectively. Prove that if $P$ is the midpoint of $B_2C_2$ then $ABC$ is isosceles.