Question 20
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 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.