Plane Geometry - Bruce E. Shapiro

214 SECTION 40. CIRCLES AND TRIANGLESHence the three perpendicular bisectors are concurrent at point O.(Uniqueness) Suppose that Γ circumscribes △ABC. By the argument in(⇐), the center of the circle occurs at the point of concurrence of the threeedge bisectors. since this point of concurrence is unique, any other circlethat circumscribes the triangle must also have a center at this same point,and be of the same radius. Hence the circumcircle is unique.Theorem 40.3 The Euclidean Parallel Postulate is equivalent ot the assertionthat every triangle can be circumscribed.This theorem says two things: (1) If the Euclidean Parallel Postulate holdsthen every triangle can be circumscribed; and (2) if the Euclidean ParallelPostulate fails, then there exists a triangle that cannot be circumscribed.We will restate and prove these statements as separate theorems 40.4 and40.6.Figure 40.1: Illustration of the proof of the Circumscribed Triangle Theorem.Theorem 40.4 If the Euclidean Parallel Postulate holds then every trianglecan be circumsribed. aa This is Euclid Book 4 Proposition 5.Proof. Assume the Euclidean Parallel Postulate and let △ABC be a triangle.Let m = ←→ AB and let k be the perpendicular bisector of AB.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012