MathNet 2021 — Part 04 Question 51

Let $ABCD$ be a trapezoid with $AB \parallel CD$, $AB=5$, $BC=9$, $CD=10$, and $DA=7$. Lines $BC$ and $DA$ intersect at point $E$. Let $M$ be the midpoint of $CD$, and let $N$ be the intersection of the circumcircles of $\triangle BMC$ and $\triangle DMA$ (other than $M$). If $EN^{2}=\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100a+b$.

Proposed by: Milan Haiman