Proof of Concurrence of the Perpendicular Bisectors of a Triangle.

By Colleen Garrett

 

Given:  Triangle ABC. Claim: The perpendicular bisectors of Triangle ABC concur at point P. Construct the perpendicular bisectors of two sides of the triangle.

Construct PB and PA and PC.  Point P is equidistant from points A and B because points lying on the perpendicular bisector are equidistant from the segments endpoints. Hence PBPA.  Similarly, PCPA. 

By transitivity, PCPB therefore, P is equidistant from points B and C.  P then lies on the perpendicular bisector of segment BC.  Hence, the perpendicular bisectors of a triangle concur at point P.