Existence of the circumcenter. Prove that the perpendicular bisectors of the sides of a triangle are concurrent. Hint: Be sure to notice that the circumcenter breaks the pattern: it is not the point of concurrence of Cevian lines. Use Exercise 5.1.4.
Start with triangle ABC and construct the perpendicular bisectors OF and OE. If a line is a perpendicular bisector of the side of a triangle, then it bisects the side into two halves and forms right angles with the side. The following can be observed from the GeoGebra construction. AE= EC and AF = FB and Angles OEA, OEC, OFA and OFB are each 90 degrees. As seen in the GeoGebra construction, perpendicular bisectors OF and OE intersect at point O. To prove that the three perpendicular bisectors of triangle ABC are concurrent, we must show that the third perpendicular bisector goes through point O as well. The Perpendicular Bisector Theorem states that if a point lies on the perpendicular bisector of a segment, it is equidistant from the endpoints of the bisected segment. Since point O lies on perpendicular bisector OD, point O is equidistant from points A and C; therefore, OA = OC. Since point O also lies on perpendicular bisector OE, it is also equidistant from points A and B; therefore, OA = OB. Using substitution, it can be concluded that OA = OB = OC. Since OC = OB, this means that point O must be equidistant from points C and B as well. Since points C and B are the endpoints of segment BC and point O is equidistant from those points, it can be concluded that point O lies on the perpendicular bisector of side BC. That perpendicular bisector has been constructed as segment OD in the GeoGebra construction. It can be concluded then that all three perpendicular bisectors, OD, OE, and OF, are concurrent at point O because point O is equidistant from all three vertices of the triangle. This point is also called the circumcenter because it is the center of the circle that circumscribes the triangle. In this GeoGebra construction, the radii of the circle are OA, OB, and OC.