Plane Geometry - Bruce E. Shapiro

202 SECTION 39. CIRCLESFigure 39.1: The segment BC is a chord of of Γ, and the lines m and l aresecants of Γ. Line n is a tangent, with point of tangency D. Line l containschord AB, which is a diameter of Γ. Points A and P are antipodel.Since the points all lie on l they are collinear and hence can be ordered.Rename them so thatA ∗ B ∗ CBecause all three points are on Γ, we haver = OA = OB = OCHence △ABO, △BCO and △ACO are all isosceles triangles with sides oflength r.By the isosceles triangle theorem (theorem 21.4), the base angles of anisosceles triangle are equal to one another, hencelet α = ∠CAO = ∠ACO (triangle △ACO)let β = ∠BAO = ∠ABO (triangle △ABO)let γ = ∠BCO = ∠CBO (triangle △BCO)By the angle sum theorem,α + α < 180 =⇒ α < 90β + β < 180 =⇒ β < 90γ + γ < 180 =⇒ γ < 90« CC BY-NC-ND 3.0. Revised: 18 Nov 2012