Question 28
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 $K \neq B$ and $N \neq C$ be the points of intersection of $BC$ with $\omega_1$ and $\omega_2$ respectively. Let also $P \neq K$ and $M \neq N$ 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.
$B \in \omega_1$, $C \in \omega_2$, and the radius of $\omega_1$ is smaller than that of $\omega_2$. Let $K \neq B$ and $N \neq C$ be the points of intersection of $BC$ with $\omega_1$ and $\omega_2$ respectively. Let also $P \neq K$ and $M \neq N$ 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.