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.