Question 24
Let $\omega_1$ and $\omega_3$ be two circles, touching externally in a common point $P$. Let further $\omega_2$ and $\omega_4$ be two circles touching externally in $P$. Suppose that for $i \in \{1, 2, 3, 4\}$ $\omega_i$ intersect $\omega_{(i \pmod 4)+1}$ again in $A_i$. Let $\ell_1$ be the common tangent of $\omega_1$ and $\omega_3$ and $\ell_2$ be the common tangent of $\omega_2$ and $\omega_4$. Show that $A_1, A_2, A_3$ and $A_4$ are con-cyclic if and only if $\ell_1$ and $\ell_2$ are orthogonal.