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 equidistant,equidistant,on two sides,equivalent from the endpoints,midpoints,endpoints of the segment, AO = CO and also AO = BO. (By transitivity,commutativity,associativity,symmetry,transitivity, 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 vertices,vertices,centers,midpoints of ΔABC. This circle is said to be the circumcircle of the triangle and its center O is known as the circumcenter.
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