Question 62
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 the segment $CD$. $CB$ is extended to meet the circle $\Gamma_2$ at $F$, while $DB$ is extended to meet the circle $\Gamma_1$ 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 $AP^2 = CA^2 + PE^2$.
a. Prove that $\angle EPF = 2\angle CAE$.
b. Prove that $AP^2 = CA^2 + PE^2$.