Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem A. 368. (March 2005)

A. 368. The interior bisector of the angle A of the triangle ABC cuts the incircle c at two points; the one closer to A is denoted by OA. Define the points OB and OC similarly on the internal bisectors of the angles B and C, respectively. The circle cA is drawn about OA and it is touching the sides AB and CA. Similarly, cB is about OB and it is touching the sides BC and AB, finally the circle cC is about OC and it is touching the sides CA and BC. Taken any two of the circles cA, cB and cC consider their common external tangents different from the corresponding sides of the triangle, respectively. Prove that these three tangents are concurrent.

(5 pont)

Deadline expired on April 15, 2005.

### Statistics:

 7 students sent a solution. 5 points: Jankó Zsuzsanna, Kiss-Tóth Christian, Mánfay Máté, Molnár 999 András, Paulin Roland, Strenner Balázs. 4 points: Pálinkás Csaba.

Problems in Mathematics of KöMaL, March 2005