#### 12. Prove that the perpendicular bisectors
of a triangle are concurrent.

####

In order to begin this proof, I constructed the segments from
the point of intersection to the vertices. These are the segments
that I am trying to prove are equal. I also constructed segments
from the midpoints to the point of intersection as well. And,
in order to speak clearly, I will label the points.

I know that:

BG=BG from the reflexive property,

AB=BC from the definition of perpendicular bisector (B is the
bisector of AC)

Angle(ABG)=Angle(CBG)=90 degrees from the definiton of perpendicular
bisector (BG is perpendicular to A.

So, by Side-Angle-Side Congruence,

By CPCTC,

#### AG = GC

Likewise,

By CPCTC,

#### GC = GE

#### GE = GA respectively

Thus,

#### AG = GC = GE.