Question 36
In triangle $A B C$, $M$ is the midpoint of $A B$ and $N$ is the midpoint of $C M$. Let $X$ be a point satisfying $\angle X M C=\angle M B C$ and $\angle X C M=\angle M C B$, such that $X$ and $B$ lie on different sides of $C M$. Let $\Omega$ be the circumcircle of triangle $A M X$.
a) Prove that $C M$ is tangent to $\Omega$.
b) Prove that the lines $N X$ and $A C$ intersect on $\Omega$.
a) Prove that $C M$ is tangent to $\Omega$.
b) Prove that the lines $N X$ and $A C$ intersect on $\Omega$.