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points defined by equal ratios on sides of triangle, area of triangle

Source: Spanish Mathematical Olympiad 1989 P2

August 2, 2018
ratioarea of a trianglegeometry

Problem Statement

Points A′,B′,C′A' ,B' ,C' on the respective sides BC,CA,ABBC,CA,AB of triangle ABCABC satisfy AC′AB=BA′BC=CB′CA=k\frac{AC' }{AB} = \frac{BA' }{BC} = \frac{CB' }{CA} = k. The lines AA′,BB′,CC′AA' ,BB' ,CC' form a triangle A1B1C1A_1B_1C_1 (possibly degenerate). Given kk and the area SS of △ABC\triangle ABC, compute the area of △A1B1C1\triangle A_1B_1C_1.
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