Plane Geometry - Bruce E. Shapiro

Section 40Circles and TrianglesDefinition 40.1 A circle that contains all three vertices of a triangle issaid to circumscribe the triangle. The circle is called the circumcircle,and its center the circumcenter of the triangle.Theorem 40.2 (Circumscribed Triangle Theorem) A triangle can becircumscribed if and only if the perpendicular bisectors of the sides of thetriangle are concurrent. If a triangle can be circumscribed, then the circumcenterand the circumcircle are unique.Proof. Let △ABC be a triangle with perpendicular bisectors l, m, and nfor segments AB, AC, and BC(⇒) We have already shown in theorem 36.2 that l, m, and n meet a singlepoint O that is equidistant from the vertices. Hence circle C(O, r) wherer = AO circumscribes the triangle.(⇐) Suppose △ABC is circumscribed by some circle Γ = C(O, r).Since Γ contains the vertices A, B, and C, we know that they are equidistantfrom O, i.e,r = AO = BO = COSince AO = BO, we know that O lies on the perpendicular bisector of AB(pointwise characterization of perpendicular bisectors, theorem 28.1).Since CO = BO, we know that O lies on the perpendicular bisector of BC(pointwise characterization of perpendicular bisectors, theorem 28.1).Since AO = CO, we know that O lies on the perpendicular bisector of AC(pointwise characterization of perpendicular bisectors, theorem 28.1).213