Many Points, Many Segments

Geometry Level 5

\(A,B,C\) are points on circle \(\omega\). Suppose points \(D,E\) are on \(AB\) and points \(F,G\) are on \(AC\) such that \[\begin{align} AD=24\\DE=4\\BE=12\\AF=15\\FG=27\\CG=18\end{align}\] If \(EG\) interesect \(\omega\) at \(X,Y\), find \[\frac {\sin \angle XDY}{\sin \angle XFY}\]

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