1967 AHSME Problems/Problem 34

Problem

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:

$\textbf{(A)}\ \frac{n^2-n+1}{(n+1)^2}\qquad \textbf{(B)}\ \frac{1}{(n+1)^2}\qquad \textbf{(C)}\ \frac{2n^2}{(n+1)^2}\qquad \textbf{(D)}\ \frac{n^2}{(n+1)^2}\qquad \textbf{(E)}\ \frac{n(n-1)}{n+1}$

Solution

WLOG, let's assume that $\triangle ABC$ is equilateral. Therefore, $[ABC]=\frac{(1+n)^2\sqrt3}{4}$ and $[DBE]=[ADF]=[EFC]=n \cdot \sin(60)/2$ . Then $[DEF]=\frac{(n^2-n+1)\sqrt3}{4}$ . Finding the ratio yields $\fbox{A}$ . -Dark_Lord

1967 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 33	Followed by Problem 35
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1967 AHSME Problems/Problem 34

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