Question 49
A circle $\gamma$ with center $I$ and radius $R$ is inscribed in quadrilateral $ABCD$. Another circle $\omega$ with center $O$, ($I \neq O$) and radius $r$ is situated inside the quadrilateral $ABCD$. Circles $\gamma_B, \gamma_C, \gamma_D, \gamma_A$ inscribed in the angles $\angle ABC, \angle BCD, \angle CDA, \angle DAB$ are tangent to the circle $\omega$ at points $B_1, C_1, D_1, A_1$ respectively. If circles $\gamma_B, \gamma_C$ tangent to $\omega$ externally and circles $\gamma_D, \gamma_A$ tangent to $\omega$ internally and $AA_1 \cap DD_1 = P, BB_1 \cap CC_1 = Q$ then find the value of $\frac{IP+IQ}{PQ}$.
