Τρίτη, 17 Οκτωβρίου 2017

2006 IMO Shortlist G7

In a triangle $ABC$, let $M_a, M_b, M_c$  be respectively the midpoints of the sides $BC, CA, AB$ and $T_a, T_b, T_c $ be the midpoints of the arcs $BC, CA, AB $ of the circumcircle of $ABC$, not containing the opposite vertices. For $ i \in \{a, b, c\}$, let $\omega_i $be the circle with $M_iT_i$ 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 $ABC$ and find the ratio of similitude.


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