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Collinear and area problem

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September 5, 2010
geometrycircumcirclegeometry unsolved

Problem Statement

Let there be given three circles K1,K2,K3K_1,K_2,K_3 with centers O1,O2,O3O_1,O_2,O_3 respectively, which meet at a common point PP. Also, let K1∩K2={P,A},K2∩K3={P,B},K3∩K1={P,C}K_1 \cap K_2 = \{P,A\}, K_2 \cap K_3 = \{P,B\}, K_3 \cap K_1 = \{P,C\}. Given an arbitrary point XX on K1K_1, join XX to AA to meet K2K_2 again in YY , and join XX to CC to meet K3K_3 again in Z.Z. (a) Show that the points Z,B,YZ,B, Y are collinear. (b) Show that the area of triangle XYZXY Z is less than or equal to 44 times the area of triangle O1O2O3.O_1O_2O_3.
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