Ukrainian Mathematical Olympiad

Non-intersecting circles ${\omega }_1$ and ${\omega }_2$ of radii are inscribed into angle $BAC$ with

$B\in {\omega }_1$, $C\in {\omega }_2$, and the radius of ${\omega }_1$ is smaller than that of ${\omega }_2$. Let $K\ne B$ and $N\ne C$ be the points of intersection of $BC$ with ${\omega }_1$ and ${\omega }_2$ respectively. Let also $P\ne K$ and $M\ne N$ be the points of intersection of $AK$ with ${\omega }_1$ and of $AN$ with ${\omega }_2$ respectively. Prove that $A$ and the circumcenters of triangles $ACM$ and $ABP$ are collinear.