IGO 2019 I3

Three circles $\omega_1$ , $\omega_2$ and $\omega_3$ pass through one common point, say $P$ . The tangent line to $\omega_1$ at $P$ intersects $\omega_2$ and $\omega_3$ for the second time at points $P_{1,2}$ and $P_{1,3}$ , respectively. Points $P_{2,1}$ , $P_{2,3}$ , $P_{3,1}$ and $P_{3,2}$ are similarly defined. Prove that the perpendicular bisector of segments $P_{1,2}P_{1,3}$ , $P_{2,1}P_{2,3}$ and $P_{3,1}P_{3,2}$ are concurrent.

Proposed by Mahdi Etesamifard