7. A point is on the perpendicular bisector of a line segment if and only if it lies the same distance from the two endpoints.

There are two things that need to be proved here. The first is that if a point is on the perpendicular bisector of a line segment, then it is equidistant from the two endpoints of the segment.

If we only use two column proofs, the student might get the idea that all proofs have to be two column proofs. This is not so. It is just that two column proofs work very well for congruent triangle proofs. In a congruent triangle proof, we first need to get the three parts of one triangle congruent to the corresponding three parts in the other triangle, note that we have congruent triangles, then conclude that the things we are trying to prove to be congruent will then be corresponding parts of the congruent triangles. That is a minimum of five steps, each step having a reason, which is a previously established statement. The two column format helps the student to keep all of these ideas straight and organized.

However, the fact of the matter is, that when we get away from congruent triangle proofs, the two column format does not always work as well. This result is an example. While it is possible to devise a two column proof, a prose proof using the isosceles triangle theorems might prove to be simpler.

If the point is on the perpendicular bisector of the line segment between the two points, then in the triangle formed by the base being the line segment, and the point being the vertex, the line from the vertex of the triangle to the midpoint of the base is perpendicular to the base, so the triangle is isosceles, and the point is equidistant from the endpoints of the line segment.

For the converse - if the point is equidistant from the endpoints of the line segment, then we again have an isosceles triangle, and the line from the vertex to the midpoint of the base will be perpendicular to the base, and thus be the perpendicular bisector of the base.