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

1. The line segment meets the base at its midpoint,
2. The line segment is perpendicular to the base.
3. 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.

1 -> 2

Given: AB = AC, BD = DC

To prove: AD is perpendicular to BC

2 -> 3

Given: AB = AC, AD is perpendicular to BC

To prove: angle BAD = angle CAD

3 -> 1

Given: AB = AC, angle BAD = angle CAD

To prove: BD = DC