Let $E,$ $F,$ $G,$ and $H$ be points on a circle such that $EF = 22$ and $GH = 81.$ Point $P$ is on segment $\overline{EF}$ with $EP = 12,$ and $Q$ is on segment $\overline{GH}$ with $GQ = 6.$ Also, $PQ = 15.$ Line segment $\overline{PQ}$ is extended to the circle at points $X$ and $Y.$ Find $XY.$
At first I thought that getting the arc length of the different sectioned arcs would help me to find the chord length by working from the arcs, but now I am unsure if that is the correct way to approach this problem with the information given.
HINT.
Intersecting chords theorem: $$ \cases{ x(15+y)=10\cdot12\\ y(15+x)=75\cdot6\\ } $$