Question 54
Let $I$ be the incentre of a non-equilateral triangle $ABC$, $I_{A}$ be the $A$-excentre, $I_{A}'$ be the reflection of $I_{A}$ in $BC$, and $l_{A}$ be the reflection of line $A I_{A}'$ in $A I$. Define points $I_{B}, I_{B}'$ and line $l_{B}$ analogously. Let $P$ be the intersection point of $l_{A}$ and $l_{B}$.
a. Prove that $P$ lies on line $OI$ where $O$ is the circumcentre of triangle $ABC$.
b. Let one of the tangents from $P$ to the incircle of triangle $ABC$ meet the circumcircle at points $X$ and $Y$. Show that $\angle X I Y = 120^{\circ}$.
a. Prove that $P$ lies on line $OI$ where $O$ is the circumcentre of triangle $ABC$.
b. Let one of the tangents from $P$ to the incircle of triangle $ABC$ meet the circumcircle at points $X$ and $Y$. Show that $\angle X I Y = 120^{\circ}$.