Question 100
Points $B$, $D$, $E$, $F$, $C$ lie on a line. Let $A$ be a point outside the line and let $\omega_1, \omega_2, \omega_3, \omega_4$ be incircles of the triangles $ABD$, $ADE$, $AEF$, $AFC$ respectively. Let $p_1, p_2, p_3, p_4$ be outwardly inscribed circles of the triangles $ABD$, $ADE$, $AEF$, $AFC$ at the vertex $A$. Prove that if circles $\omega_1, \omega_2, \omega_3, \omega_4$ have common tangent different from $BC$ then circles $p_1, p_2, p_3, p_4$ also have common tangent different from $BC$.