$ABCD$ is a cyclic quadrilateral with $AB=11$ and $CD= 19$. Points $P$ and $Q$ are on $AB$ and $CD$ respectively such that $AP=6,\, BP=5\, DQ=7$ and $CQ=12,\, PQ=27$. Extend $PQ$ till it meets the circle at point $R$ and point $S$.
This is what I have so far:
- I constructed chords $DS,\, CS,\, AD,\, BC,\, AR,\, RB$ and $BC$
- Ptolemy in ABCD $AB\times DC+ AD \times BC= AC \times BD$
- I think Ptolemy needs to be done multiple times in all the cyclic quads i just don't know where to start or how to do that
Thank you so much!