« Vrati se
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.

Slični zadaci

#NaslovOznakeRj.KvalitetaTežina
2168IMO Shortlist 2004 problem G75
2169IMO Shortlist 2004 problem G815
2176IMO Shortlist 2004 problem N70
2221IMO Shortlist 2006 problem G56
2225IMO Shortlist 2006 problem G95
2256IMO Shortlist 2007 problem G85