MathNet 2021 — Part 07 Question 20

In triangle $ABC$, let $P$ and $Q$ be points on side $BC$ and suppose that the orthocenter of triangle $ACP$ and the orthocenter of triangle $ABQ$ coincide. Given that $AB = 10$, $AC = 11$, $BP = 5$, $CQ = 6$, find the length of $BC$.