Problem 892

Let $\omega_{1}$ and $\omega_{2}$ be two circles, $t$ and $t^{\prime}$ two tangents to the circles. We denote $T_{1}$ and $T_{2}$ the points of tangency of $t$ with $\omega_{1}$ and $\omega_{2}$ respectively, and $T_{1}^{\prime}$ and $T_{2}^{\prime}$ the points of tangency with $t^{\prime}$. Let $M$ be the midpoint of $\left[T_{1} T_{2}\right]$, $P_{1}$ and $P_{2}$ the intersections of $\left(M T_{1}^{\prime}\right)$ and $\left(M T_{2}^{\prime}\right)$ with $\omega_{1}$ and $\omega_{2}$ respectively. Show that the quadrilateral $P_{1} P_{2} T_{1}^{\prime} T_{2}^{\prime}$ is cyclic.