Question

In the construction of the circumcenter, why do you only need to construct two perpendicular bisectors, and not all three?

Answer 1

We know that the Circumcenter is the intersection of a triangle's right (perpendicular) bisectors. Once you have found two of these you can determine their point of intersection. 

It is a proven mathematical concept that the third perpendicular bisector will also cross through this intersection point. While it is not necessary to construct all three it is often done so to verify that the intersection of the first two perpendicular bisectors was in fact correct.