Question 65
Let $M$ and $N$ be arbitrary points on the side $AB$ of a triangle $ABC$ such that $M$ lies between $A$ and $N$. The line through $M$ parallel to $AC$ meets the circumcircle of $\triangle MNC$ at point $P$, and the line through $M$ parallel to $NC$ meets the circumcircle of $\triangle AMC$ at point $Q$. Analogously, the line through $N$ parallel to $BC$ meets the circumcircle of $\triangle MNC$ at point $K$ and the line through $N$ parallel to $MC$ meets the circumcircle of $\triangle BNC$ at point $L$. Prove that:
a) the points $P$, $Q$ and $C$ are collinear;
b) the points $P$, $Q$, $K$ and $L$ are concyclic if and only if $AM = BN$.
a) the points $P$, $Q$ and $C$ are collinear;
b) the points $P$, $Q$, $K$ and $L$ are concyclic if and only if $AM = BN$.