Question 153
Let $ABC$ be a triangle with $AB \lt AC \lt BC$, incentre $I$ and incircle $\omega$. Let $X$ be the point in the interior of side $BC$ such that the line through $X$ parallel to $AC$ is tangent to $\omega$. Similarly, let $Y$ be the point in the interior of side $BC$ such that the line through $Y$ parallel to $AB$ is tangent to $\omega$. Let $AI$ intersect the circumcircle of triangle $ABC$ again at $P \neq A$. Let $K$ and $L$ be the midpoints of $AB$ and $AC$, respectively.
Prove that $\angle KIL + \angle YPX = 180^{\circ}$.
(Poland)
Prove that $\angle KIL + \angle YPX = 180^{\circ}$.
(Poland)