Problem A. 540. (September 2011)
A. 540. Let A_{1}A_{2}A_{3} be a nonequilateral triangle, let point M be its orthocenter, let F be its Feuerbach point, and let k be the circumcircle of the triangle. For i=1,2,3 denote by k_{i} the circle that is internally tangent to k and tangent to the sides A_{i}A_{i+1} and A_{i}A_{i+2}. (The indices are considered modulo 3, i.e. A_{4}=A_{1} and A_{5}=A_{2}.) Let T_{i} be the point of tangency between k and k_{i}. Prove that the lines A_{1}T_{1}, A_{2}T_{2}, A_{3}T_{3} and MF are concurrent.
Proposed by: Gábor Damásdi and Márton Mester, Budapest
(5 pont)
Deadline expired on 10 October 2011.
Solution. Denote by b the incircle and by f the 9point circle and let H be the external homothety center between f and k.
Apply Monge's circle theorem to the circles b, k and k_{i}. The three external homothety centers between b and k_{i}, k and k_{i}, b and k are A_{i}, T_{i} and H, respectively; by the theorem these points are collinear. Therefore the line A_{i}T_{i} passes through H (i=1,2,3).
Now apply Monge's theorem to the circles b, f and k. The three homotety centers between them are M, F and H. Therefore the line FM also passes through H.
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