Hi Elizabeth.

There's no mystery at all!

When you connect a corner of an equilateral triangle to the midpoint on the opposite side there are two important things that happen: (a) The angle is bisected; it is split into two equal sized angles and (b) the intersection at the midpoint on the opposite side is perpendicular, so you have created two identical (mirror image) right triangles.

In fact, the same can be said about isosceles triangles and the angle shared by the two common-length sides. It works for any angle in an equilateral triangle.

Now, what does the Pythagorean theorem tell you about the altitude (the length of that line you drew) and the other sides?

Cheers,
Stephen La Rocque