Author's Note: I got this puzzle from Nicolay Tchernitsky.
Examine the drawing in Figure 1.
Figure 1: Equilateral Triangles Puzzle
The triangle ABC is an equilateral triangle with an area S and a side length a. The line CF is a continuation of the line AC, AD is a continuation of BA and BE is a continuation of CB. The length of all continuation segments (CF, AD and BE) is a—the same as the length of triangle ABC’s sides. The puzzle is to calculate the area of the triangle DEF.