Question 60
Let $ABCD$ be a quadrilateral with equal angles $B$ and $D$. Circles $\omega_1, \omega_2$ are symmetric with respect to $AC$; $\omega_1$ passes through $B$ and intersects $AB$ and $BC$ for the second time at $K$ and $L$, respectively, and $\omega_2$ passes through $D$ and intersects $CD$ and $AD$ for the second time at $M$ and $N$, respectively.
Prove that $KM$ and $LN$ intersect on $AC$.
Prove that $KM$ and $LN$ intersect on $AC$.