Let $\Gamma_1$ and $\Gamma_2$ be two circles with different radii, with $\Gamma_1$ the smaller one. The two circles meet at distinct points $A$ and $B$. $C$ and $D$ are two points on the circles $\Gamma_1$ and $\Gamma_2$, respectively, and such that $A$ is the midpoint of $CD$. $CB$ is extended to meet $\Gamma_2$ at $F$, while $DB$ is extended to meet $\Gamma_1$ at $E$. The perpendicular bisector of $CD$ and the perpendicular bisector of $EF$ meet at $P$. (a) Prove that $\angle{EPF} = 2\angle{CAE}$. (b) Prove that $AP^2 = CA^2 + PE^2$.