Question 53
Three circles $\omega_1$, $\omega_2$ and $\omega_3$ with non-colinear centres $O_1$, $O_2$ and $O_3$ are drawn such that $\omega_1$ externally touches $\omega_2$ and $\omega_3$ at the points $P$ and $Q$ respectively. An arbitrary point $C$ is chosen on $\omega_1$. The line $CP$ intersects $\omega_2$ for the second time at the point $B$, and the line $CQ$ intersects $\omega_3$ for the second time at the point $A$. Point $O$ is the circumcenter of the triangle $ABC$.
Prove that while point $C$ varies over all positions on $\omega_1$, the locus of the points $O$ is the circle and the center of this circle lies on the circumcircle of the triangle $O_1O_2O_3$. (Mikhail Karpuk)
Prove that while point $C$ varies over all positions on $\omega_1$, the locus of the points $O$ is the circle and the center of this circle lies on the circumcircle of the triangle $O_1O_2O_3$. (Mikhail Karpuk)