Let $A B C$ be a triangle. The circle $\omega_{A}$ through $A$ is tangent to line $B C$ at $B$. The circle $\omega_{C}$ through $C$ is tangent to line $A B$ at $B$. Let $\omega_{A}$ and $\omega_{C}$ meet again at $D$. Let $M$ be the midpoint of line segment [ $B C]$, and let $E$ be the intersection of lines $M D$ and $A C$. Show that $E$ lies on $\omega_{A}$.