Question 165
Let $ABCD$ be a trapezoid with $(AB)$ parallel to $(CD)$. Suppose that there are two circles $\omega_{1}$ and $\omega_{2}$ inside the trapezoid such that $\omega_{1}$ is tangent to the sides $[DA]$, $[AB]$ and $[BC]$, and $\omega_{2}$ is tangent to the sides $[BC]$, $[CD]$ and $[DA]$. Let $(d_{1})$ be the second tangent (other than $(AD)$) to $\omega_{2}$ through $A$, and let $(d_{2})$ be the second tangent (other than $(BC)$) to $\omega_{1}$ through $C$.
Prove that $(d_{1})$ and $(d_{2})$ are parallel.
Prove that $(d_{1})$ and $(d_{2})$ are parallel.