### IMO Shortlist 2006 problem G7

Kvaliteta:
Avg: 0,0
Težina:
Avg: 9,0
Dodao/la: arhiva
2. travnja 2012.
In a triangle $ABC$, let $M_{a}$, $M_{b}$, $M_{c}$ be the midpoints of the sides $BC$, $CA$, $AB$, respectively, and $T_{a}$, $T_{b}$, $T_{c}$ be the midpoints of the arcs $BC$, $CA$, $AB$ of the circumcircle of $ABC$, not containing the vertices $A$, $B$, $C$, respectively. For $i \in \left\{a, b, c\right\}$, let $w_{i}$ be the circle with $M_{i}T_{i}$ as diameter. Let $p_{i}$ be the common external common tangent to the circles $w_{j}$ and $w_{k}$ (for all $\left\{i, j, k\right\}= \left\{a, b, c\right\}$) such that $w_{i}$ lies on the opposite side of $p_{i}$ than $w_{j}$ and $w_{k}$ do.
Prove that the lines $p_{a}$, $p_{b}$, $p_{c}$ form a triangle similar to $ABC$ and find the ratio of similitude.
Izvor: Međunarodna matematička olimpijada, shortlist 2006