Sarah,

This observation can be grouped with a set of connected properties - if you think about symmetry of the triangle.
What is being said, if you work though this a bit, is that the altitude works as a mirror to take the triangle onto itself.
- the property of the altitude being through the midpoint means that a reflection in that mirror takes the base onto itself. The 'top' vertex is already on the mirror so it reflects to itself.

Once you have a mirror - you get a package of properties involving angles, the other sides etc. which are grouped into one of concepts (definitions) used to describe triangles with all of these properties.

Walter Whiteley
York University