IMO Shortlist 2006 problem G7

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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