4. To drop a perpendicular from a point not on a line to the line.

In the construction for dropping a perpendicular from a point to a line, we are

This proof is more involved. While we know that triangle ABC is isosceles, we will need to know that AD either hits the base at its midpoint, is perpendicular to the base, or bisects the vertex angle before we can use the Isosceles Triangle Theorems. That is is perpendicular to the base is what we have to prove. The one we will be able to establish is that it bisects the vertex angle with the congruent triangle proof above.