Question 21
In a triangle $ABC$, $\hat{A} = 2\hat{C}$ and $2\hat{B} = \hat{A} + \hat{C}$. The bisector of angle $\hat{C}$ meets side $AB$ at $E$, and $F$ is the midpoint of segment $AE$. The altitude to side $BC$ is $AD$. The perpendicular bisector of segment $DF$ meets side $AC$ at $M$.
Prove that $AM = CM$.
Prove that $AM = CM$.