JBMO 2001 Problem 3

Let $ABC$ be an equilateral triangle and $D$ , $E$ points on the sides $[AB]$ and $[AC]$ respectively. If $DF$ , $EF$ (with $F\in AE$ , $G\in AD$ ) are the interior angle bisectors of the angles of the triangle $ADE$ , prove that the sum of the areas of the triangles $DEF$ and $DEG$ is at most equal with the area of the triangle $ABC$ . When does the equality hold?

Greece