Question 65
Let $ABCD$ be a quadrilateral having no two parallel sides, inscribed into the circle $\Omega$. Let $\omega_a, \omega_b, \omega_c, \omega_d$ be circles inscribed into triangles $DAB, ABC, BCD, CDA$, respectively. Let us draw common external tangents $t_1, t_2, t_3, t_4$ to pairs of circles $\omega_a$ and $\omega_b$, $\omega_b$ and $\omega_c$, $\omega_c$ and $\omega_d$, $\omega_d$ and $\omega_a$, respectively, so that lines $t_1, t_2, t_3, t_4$ do not contain the sides of $ABCD$. The quadrilateral whose consecutive sides lie on $t_1, t_2, t_3, t_4$ (in this order) is inscribed into the circle $\Gamma$. Prove that the three lines connecting centers of $\omega_a$ and $\omega_c$, $\omega_b$ and $\omega_d$, $\Omega$ and $\Gamma$, are concurrent.