2020 HMMT November General Round Problem 7

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.)

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2020 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.