Mathematical and Physical Journal
for High Schools
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Problem A. 389. (January 2006)

A. 389. Point P lies in the interior of the acute triangle ABC. The circumcenters of triangles ABC, BCP, CAP and ABP are O, A1, B1 and C1, respectively. Prove that

\frac{\mathrm{area}\,(\Delta A_1B_1O)}{\mathrm{area}\,(\Delta ABP)} = \frac{\mathrm{area}\,(\Delta B_1C_1O)}{\mathrm{area}\,(\Delta BCP)} = \frac{\mathrm{area}\,(\Delta C_1A_1O)}{\mathrm{area}\,(\Delta CAP)}.

(5 pont)

Deadline expired on February 15, 2006.

Statistics:

13 students sent a solution.
5 points:Bogár 560 Péter, Dücső Márton, Estélyi István, Gyenizse Gergő, Hujter Bálint, Jankó Zsuzsanna, Kisfaludi-Bak Sándor, Kónya 495 Gábor, Nagy 224 Csaba, Paulin Roland, Tomon István.
2 points:1 student.
0 point:1 student.

Problems in Mathematics of KöMaL, January 2006