Question

Write up a formal proof that the perpendicular bisectors of a triangle are concurrent, and that the point of concurrency (the circumcenter) is equidistant from all three vertices

Answer 1

The circumcenter is equidistant from the three vertices of the triangle. From the figure shown, we will prove DA = DB = DC.

1) Triangle ABC; Perpendicular bisectors of each side(Given)

2) DA = DB, DC = DB(If a point is on the perp. bisector of a segment, it is equidistant from each endpoint of the segment.)

3) DA = DB(Substitution)\

4) D is on the perpendicular bisector of seg. AB(Converse of 2)

5)DA = DB = DC, so D is equidistant from the 3 vertices of the triangle(Steps 2 and 3 and definition of equidistant)