Solution to the triangle problem

If we draw a line through the point in the center that is parallel to one side, then the problem is exactly the same as showing that the 2 perpendiculars in the top triangle sum up to the length of a perpendicular for that top triangle. If we draw a line through the point that is parallel to one of the original sides and consider only the picture for the top triangle, then it is clear that the perpendicular for the smallest triangle is a perpendicular from the vertex. Since we are dealing with equilateral triangles, if we rotate this small triangle, we see that the perpendicular added to the other perpendicular in the top triangle will add to the perpendicular for the top triangle, and then when we add the three together we get the result.
                   /  \
                  /    \
                 /      \
                /        \
               /          \
             /              \
            /                \
           /                  \
          /                    \