Question 34
Let $\triangle ABC$ be an acute-angled triangle with $AB \lt AC$, inscribed in to the circle $c(O, R)$. The extension of the altitude $AD$ intersects the circumcircle $c(O, R)$ at $E$ and the perpendicular bisector ($\mu$) of the side $AB$ meets $AD$ at $L$. The line $BL$ meets $AC$ at $M$ and the circumcircle $c(O, R)$ at $N$. Finally, $EN$ meets ($\mu$) at $Z$. Prove that:
$$
MZ \perp BC \Leftrightarrow (CA = CB \text{ or } Z \equiv O).
$$
$$
MZ \perp BC \Leftrightarrow (CA = CB \text{ or } Z \equiv O).
$$