Question 125
Triangle $ABC$ is not equilateral, $I$ is its incentre, $I_A$ is the excentre inside angle $A$, $I'_A$ is the reflection of $I_A$ in the line $BC$, and $\ell_A$ is the reflection of the line $AI'_A$ in $AI$. Define $I_B, I'_B$ and $\ell_B$ analogously. Let $P$ be the intersection of $\ell_A$ and $\ell_B$.
a. Prove that if $O$ is the circumcentre of triangle $ABC$, then $P$ lies on the line $OI$.
b. Suppose a tangent from $P$ to the incircle of triangle $ABC$ meets the circumcircle of $ABC$ at two points $X, Y$. Prove that $\angle XIY = 120^\circ$.
a. Prove that if $O$ is the circumcentre of triangle $ABC$, then $P$ lies on the line $OI$.
b. Suppose a tangent from $P$ to the incircle of triangle $ABC$ meets the circumcircle of $ABC$ at two points $X, Y$. Prove that $\angle XIY = 120^\circ$.