Mathematical and Physical Journal
for High Schools
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# Problem A. 514. (September 2010)

A. 514. There are given three circles in the plane, k0, k1 and k2, being externally tangent to each other. The center of k0 is O and one of its diameters is A1A2. Denote by B the point of tangency between k1 and k2, by C1 the point of tangency between k0 and k1 and by C2 the point of tangency between k0 and k2. The line segments A1C2 and A2C1 meet at point D in the interior of the circle k0. Let t1 and t2 be the tangent lines to the circle k0 at A1 and A2, respectively. Prove that if t1 is tangent to k1 and t2 is tangent to k2 then the line segment OB passes through point D.

(5 pont)

Deadline expired on October 11, 2010.

### Statistics:

 21 students sent a solution. 5 points: Ágoston Tamás, Backhausz Tibor, Bágyoni-Szabó Attila, Damásdi Gábor, Frankl Nóra, Gyarmati Máté, Janzer Olivér, Kalina Kende, Kiss 986 Mariann, Kovács Márton, Lenger Dániel, Mester Márton, Nagy 235 János, Nagy 648 Donát, Nagy Balázs, Strenner Péter, Szabó 928 Attila, Weisz Ágoston. 0 point: 3 students.

Problems in Mathematics of KöMaL, September 2010