2006 IMO Shortlist G7

In a triangle $A B C$, let $M_{a}, M_{b}, M_{c}$ be respectively the midpoints of the sides $B C, C A$, $A B$ and $T_{a}, T_{b}, T_{c}$ be the midpoints of the $\operatorname{arcs} B C, C A, A B$ of the circumcircle of $A B C$, not containing the opposite vertices. For $i \in\{a, b, c\}$, let $\omega_{i}$ be the circle with $M_{i} T_{i}$ as diameter. Let $p_{i}$ be the common external tangent to $\omega_{j}, \omega_{k}(\{i, j, k\}=\{a, b, c\})$ such that $\omega_{i}$ lies on the opposite side of $p_{i}$ than $\omega_{j}, \omega_{k}$ do. Prove that the lines $p_{a}, p_{b}, p_{c}$ form a triangle similar to $A B C$ and find the ratio of similitude. (Slovakia)

在三角形$A B C$中，设$M_{a}、M_{b}、M_{c}$分别为边$B C、C A$、$A B$的中点，$T_{a}、T_{b}、T_{c}$为$A B C$外接圆的$\operatorname{arcs} B C、C A、A B$的中点，不包含对边顶点。对于$i \in\{a, b, c\}$，令$\omega_{i}$ 为以$M_{i} T_{i}$ 为直径的圆。令 $p_{i}$ 为 $\omega_{j}, \omega_{k}(\{i, j, k\}=\{a, b, c\})$ 的公外切线，使得 $\omega_{i}$ 位于 $p_{i}$ 的相对侧，而 $\omega_{j}, \omega_{k}$ 则位于 $p_{i}$ 的对侧。证明线 $p_{a}、p_{b}、p_{c}$ 形成一个与 $A B C$ 相似的三角形，并求相似比。 (斯洛伐克)