Question 54
Let $ABCD$ be a parallelogram. Consider circles $\omega_1$ and $\omega_2$ such that $\omega_1$ is tangent to segments $AB$, $AD$ and $\omega_2$ is tangent to segments $BC$, $CD$. Suppose that there exists a circle tangent to lines $AD$, $DC$ and externally tangent to $\omega_1$, $\omega_2$. Prove that there exists a circle tangent to lines $AB$ and $BC$ and externally tangent to $\omega_1$, $\omega_2$.