Question 93
Point $P$ is located inside a square $A B C D$ of side length $10$. Let $O_{1}, O_{2}, O_{3}, O_{4}$ be the circumcenters of $P A B$, $P B C$, $P C D$, and $P D A$, respectively. Given that $P A + P B + P C + P D = 23 \sqrt{2}$ and the area of $O_{1} O_{2} O_{3} O_{4}$ is $50$, the second largest of the lengths $O_{1} O_{2}, O_{2} O_{3}, O_{3} O_{4}, O_{4} O_{1}$ can be written as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.