Plane Geometry - Bruce E. Shapiro

Section 36Triangle CentersDefinition 36.1 Three distinct lines are concurrent at point P if P lieson all three lines, and P is called the point of concurrency of the threelines.Theorem 36.2 The three perpendicular bisectors of the sides of any trianglemeet at a single point we call the circumcenter that is equidistantfrom the vertices of the triangle.Proof. Let △ABC be a triangle. Denote the midpoint of AB by P , themidpoint of BC by Q, and the midpoint of CA by R.Let l be the bisector of AB and m the bisector of BC. Then either l ‖ m,l = m, or l intersects m.Suppose l ‖ m. Then since AB ⊥ l, this means AB ⊥ m. Since we alsohave BC ⊥ m this means that either BC ‖ AB or ←→ BC = ←→ AB. Since BCand AB are two distinct sides of the same triangle, neither case is possible.Hence l ̸‖ m.Similarly, if l = m, either BC ‖ AB or ←→ BC = ←→ AB, which we have just saidis impossible. Hence l ≠ m.Therefore l must intersect m. Define the point O = l ∩ m. Then l = ←→ P Oand m = ←→ QO by definition of m and l.By SAS, △AOP ∼ = △BOP , hence AO = BO.Also by SAS, △BOQ ∼ = △COQ, hence BO = CO.Since R is the midpoint of AC, then by SSS, △AOR ∼ = △COR.181