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ratio of triangle areas equal to ratio of chords, intersecting circles related

Source: Greece JBMO TST 2003 p4

June 18, 2019
ratiocirclesareasarea of a trianglegeometry

Problem Statement

Given are two points B,CB,C. Consider point AA not lying on the line BCBC and draw the circles C1(K1,R1)C_1(K_1,R_1) (with center K1K_1 and radius R1R_1) and C2(K2,R2)C_2(K_2,R_2) with chord AB,ACAB, AC respectively such that their centers lie on the interior of the triangle ABCABC and also R1⋅AC=R2⋅ABR_1 \cdot AC= R_2 \cdot AB. Let TT be the intersection point of the two circles, different from AA, and M be a random pointof line ATAT, prove that TC⋅S(MBT)=TB⋅S(MCT)TC \cdot S_{(MBT)}=TB \cdot S_{(MCT)}
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