3. In an isosceles triangle, if a line segment goes from the vertex angle to the base, the following conditions are equivalent:

- The line segment meets the base at its midpoint,
- The line segment is perpendicular to the base.
- The line segment bisects the vertex angle.

There are actually six theorems here: 1 -> 2, 1 -> 3, 2 -> 3, 2 ->1, 3 -> 1, and 3 ->2. We can reduce the problem to proving only 3 if we do it in a circular fashion: 1 -> 2, 2 -> 3, and 3 -> 1. This will suffice, because if we wanted to establish, say, 2 -> 1, that would follow from 2 -> 3 and 3 -> 1. However, if we use that method, the student will not get to experience as many proofs, so we will prove 1-> 2 and 3, 2 -> 1 and 3, and 3 -> 1 and 2.