Putnam 1962 A3

In a triangle $ABC$ , let $A'$ be a point on the segment $BC$ , $B'$ be a point on the segment $CA$ and $C'$ a point on the segment $AB$ such that $$\frac{AB'}{B'C}= \frac{BC'}{C'A} =\frac{CA'}{A'B}=k,$$ where $k$ is a positive constant. Let $\triangle$ be the triangle formed by the interesctions of $AA'$ , $BB'$ and $CC'$ . Prove that the areas of $\triangle$ and $ABC$ are in the ratio $$\frac{(k-1)^{2}}{k^2 +k+1}.$$