In the triangle ABC, DE is the middle line. The area of the triangle CDE is 20. Find the area of the triangle ABC.

We know that the area of a triangle is half the product of its base and height.

S Δ BDE = 1/2 DE · BK, where ВK – height Δ BDE.

S Δ CDE = 1/2 DE · CN, where CN is the height of Δ CDE.

But BK = KH = CN, since DE is the middle line of Δ ABC.

Therefore, S Δ BDE = S Δ CDE = 20.

Δ BDE ≈ Δ BAC in two angles (angle B – common, angle BDE = angle A as corresponding for DE // AC and secant AB.)

And the ratio of the areas of such figures is equal to the square of the similarity coefficient.

By the property of the middle line of the triangle, k = AC / DE = 2.

So, S Δ ABC / S Δ BDE = 4;

S Δ ABC = 4 S Δ BDE = 4 20 = 80.

Answer. S Δ ABC = 80.