Question 151
Let $\omega_1$, $\omega_2$, and $\omega_3$ be three circles passing through the origin $O$ of the coordinate plane but not tangent to each other or to either axis. Denote by $(x_i, 0)$ and $(0, y_i)$, $1 \le i \le 3$, the respective intersections (besides $O$) of circle $\omega_i$ with the $x$ and $y$ axes. Prove that $\omega_1$, $\omega_2$, and $\omega_3$ have a common point $P \neq O$ if and only if the points $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are collinear.