Question 134
Let $\omega$ and $\Omega$ be the incircle and the circumcircle of an equilateral triangle $ABC$, respectively. Points $P$ and $Q$ are chosen on the sides $AC$ and $AB$, respectively, so that the segment $PQ$ contains the center of the triangle $ABC$. Circles $\Omega_b$ and $\Omega_c$ have diameters $BP$ and $CQ$, respectively. Prove that the circles $\Omega_b$ and $\Omega_c$ meet at two points one of which lies on $\omega$ and the other one lies on $\Omega$.