Mathematical and Physical Journal
for High Schools
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# 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 December 10, 2009.

### Statistics:

 8 students sent a solution. 5 points: Ágoston Tamás, Bodor Bertalan, Éles András, Frankl Nóra, Nagy 235 János, Nagy 648 Donát, Szabó 928 Attila, Weisz Ágoston.

Problems in Mathematics of KöMaL, November 2009