Question 171
Consider parallelogram $ABCD$ with $AB \gt BC$. Point $E$ on $\overline{AB}$ and point $F$ on $\overline{CD}$ are marked such that there exists a circle $\omega_1$ passing through $A, D, E, F$ and a circle $\omega_2$ passing through $B, C, E, F$. If $\omega_1, \omega_2$ partition $\overline{BD}$ into segments $\overline{BX}, \overline{XY}, \overline{YD}$ in that order, with lengths $200, 9, 80$, respectively, compute $BC$.