MathNet 2023-part-07 — 187 problems (EN–VI) Question 69

Let $ABCD$ be a square of side length $5$. A circle passing through $A$ is tangent to segment $CD$ at $T$ and meets $AB$ and $AD$ again at $X \neq A$ and $Y \neq A$, respectively. Given that $XY = 6$, compute $AT$.