HMMT 2022 G5

Let triangle $A B C$ be such that $A B=A C=22$ and $B C=11$. Point $D$ is chosen in the interior of the triangle such that $A D=19$ and $\angle A B D+\angle A C D=90^{\circ}$. The value of $B D^{2}+C D^{2}$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.