On the given triangle ABC, we construct a rectangle with equal base to the one of the sides of the triangle and the parallel side to the base passing for the midpoints of AB and AC. We construct the segment GA perpendicular to the HE.
We must show that the two triangles in the given triangle are congruent to the dotted triangles of the rectangle. In fact this happens:
1.To show congruent triangles AGH and BDH observe:
* sides AH and BH are congruent since H is the midpoint of AB
* angles AGH and BDH are right angles
* angles AHG and BHD are congruentes since they are vertical angles formed at H by AB and DF.
2.Analogous reasoning reveals also that the triangles ACT and CFE are congruent.
Thus we can conclude that the parts that compose the ABC triangle also fit perfectly in the constructed rectangle.