MathNet 2020 — Math Olympiad Vault — Part 05 Question 8

Let $ABCD$ be a square and let $X$ be any point on side $BC$ between $B$ and $C$. Let $Y$ be the point on line $CD$ such that $BX = YD$ and $D$ is between $C$ and $Y$. Prove that the midpoint of $XY$ lies on diagonal $BD$.