MathNet 2024-part-01 — 168 problems (EN–VI) Question 115

Let $ABC$ be a triangle with incenter $I$. Suppose the reflection of $AB$ across $CI$ and the reflection of $AC$ across $BI$ intersect at a point $X$. Prove that $XI$ is perpendicular to $BC$.

(The incenter is the point where the three angle bisectors meet.)