MathNet 2020 — Math Olympiad Vault — Part 03 Question 13

In triangle $ABC$ with $AB=8$ and $AC=10$, the incenter $I$ is reflected across side $AB$ to point $X$ and across side $AC$ to point $Y$. Given that segment $XY$ bisects $AI$, compute $BC^{2}$. (The incenter $I$ is the center of the inscribed circle of triangle $ABC$.)

Proposed by: Carl Schildkraut