Problem A. 491. (November 2009)
A. 491. In the triangle \(\displaystyle A_1A_2A_3\), for each \(\displaystyle i=1,2,3\), the excircle, which is tangent to the side \(\displaystyle A_{i+1}A_{i+2}\), touches the half lines \(\displaystyle A_iA_{i+1}\) and \(\displaystyle A_iA_{i+2}\) at \(\displaystyle P_i\) and \(\displaystyle Q_i\), respectively. (The indices are considered modulo 3, e.g. \(\displaystyle A_4=A_1\) and \(\displaystyle A_5=A_2\).) The lines \(\displaystyle P_iP_{i+1}\) and \(\displaystyle Q_iQ_{i+2}\) meet at \(\displaystyle R_i\); finally, the lines \(\displaystyle P_{i+1}P_{i+2}\) and \(\displaystyle Q_{i+1}Q_{i+2}\) meet at \(\displaystyle S_i\) (\(\displaystyle i=1,2,3\)). Prove that the lines \(\displaystyle R_1S_1\), \(\displaystyle R_2S_2\) and \(\displaystyle R_3S_3\) are concurrent.
(From the idea of Bálint Bíró, Eger)
(5 pont)
Deadline expired on 10 December 2009.
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