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$$E$$ is a point on the side $$AC$$ of a triangle $$ABC$$. $$D$$ and $$F$$ are points on the sides $$BC$$ and $$AB$$ respectively such that the line segments $$DE$$, $$FE$$ are parallel to the sides $$AB$$ and $$BC$$ respectively. If $$\Delta_1$$, $$\Delta_2$$, $$\Delta_3$$ are the areas of the quadrilateral $$BDEF$$, triangle $$AEF$$, triangle $$CDE$$ respectively then evaluate $$\frac{\Delta_1^2}{\Delta_2 \Delta_3}$$.
Correct Answer: 4
Let $$\Delta$$ be the area of $$ABC$$ and $$\frac{AE}{AC} = k$$. Triangle $$AEF$$ is similar to $$ABC$$ with ratio $$k$$, and triangle $$EDC$$ is similar to it with ratio $$1 - k$$, so $$\Delta_2 = k^2\Delta$$ and $$\Delta_3 = (1 - k)^2\Delta$$. Then $$\Delta_1 = \Delta\left[1 - k^2 - (1 - k)^2\right] = 2k(1 - k)\Delta$$. So $$\frac{\Delta_1^2}{\Delta_2\Delta_3} = \frac{4k^2(1 - k)^2}{k^2(1 - k)^2} = 4$$.
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