smc

Question

Points D, E, F are taken respectively on sides AB, BC, and CA of triangle ABC so that AD:DB=BE:CE=CF:FA=1:n. The ratio of the area of triangle DEF to that of triangle ABC is:

Accepted Answer

WLOG, let's assume that ABC is equilateral. Therefore, [ABC]=(1+n)^234 and [DBE]=[ADF]=[EFC]=n (60)/2. Then [DEF]=(n^2-n+1)34. Finding the ratio yields n^2-n+1(n+1)^2.