2006 IMO Shortlist G6

Circles $\omega_{1}$ and $\omega_{2}$ with centres $O_{1}$ and $O_{2}$ are externally tangent at point $D$ and internally tangent to a circle $\omega$ at points $E$ and $F$, respectively. Line $t$ is the common tangent of $\omega_{1}$ and $\omega_{2}$ at $D$. Let $A B$ be the diameter of $\omega$ perpendicular to $t$, so that $A, E$ and $O_{1}$ are on the same side of $t$. Prove that lines $A O_{1}, B O_{2}, E F$ and $t$ are concurrent. (Brasil)

圆 $\omega_{1}$ 和 $\omega_{2}$ 的圆心为 $O_{1}$ 和 $O_{2}$，分别在点 $D$ 处与圆 $\omega$ 外切，在点 $E$ 和 $F$ 处与圆 $\omega$ 内切。线$t$是$\omega_{1}$和$\omega_{2}$在$D$处的公切线。设$A B$为垂直于$t$的$\omega$的直径，使得$A、E$和$O_{1}$位于$t$的同一侧。证明 $A O_{1}、B O_{2}、E F$ 和 $t$ 行是并发的。 (巴西)