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