Question 101
A pentagon $ABCDE$ is inscribed in a circle $\Omega$, and it satisfies $AC = AD$ and $BC \parallel DE$. Take a point $P$ on the arc $CD$ of $\Omega$ which does not contain $A$. Let $P_1, P_2, P_3, P_4$, and $P_5$ be the points symmetric to $P$ with respect to the lines $AB, BC, CD, DE$, and $EA$, respectively. When $PC : PD = P_1P_2 : P_4P_5 = 2 : 3$ and $CD : P_2P_4 = 4\sqrt{2} : 11$, determine the value of $\frac{P_1P_3}{P_1P_5}$.