Outline Mathematics
Geometry

Existence of the Circumcenter

This is Euclid IV.5.

Bisect the straight lines AB and AC at the points N and M. Draw the perpendicular bisectors to AB and AC. (The two may meet within the triangle ABC, or on the side BC, or outside BC.)

Let O be the point of intersection. Join O to A, B, C. Since the perpendicular bisector of a line segment is the locus of points from the of the segment, AO = CO and also AO = BO. (By , BO = CO.) The three distances are equal and each can be taken as the radius of the circle centered at O. Such a circle, passes through all three of DABC. This circle is said to be the circumcircle of the triangle and its center O is known as the circumcenter.