3. To erect a perpendicular from a point
In the construction for erecting a perpendicular from a point, we are given
From the construction we are given
Note that up until the last line, this is just the construction for bisecting an angle. We have the same "Givens" and the first lines of the proof are the same. If we bisect the straight angle at A, we will get a right angle. However, it is possible to prove this using the Isosceles Triangle Theorems.
Note that the first proof simply reiterates the proof of the Isosceles Triangle Theorem. Since we have already proved it, we don't have to prove it again. We need simply refer to it.