Plane Geometry - Bruce E. Shapiro

Section 41Euclidean CirclesIn this section we assume that the Euclidean Parallel Postulate holds forall theorems.Theorem 41.1 Let △ABC be a triangle and let M be the midpoint ofAB. If AM = MC then ∠ACB is a right angle.Proof. See figure 41.1. Since AM = MC, by the isoceles triangle theoremα = β.Since BM = MC, them by the isosceles triangle theorem δ = γ.Henceσ(△ABC) = α + (β + γ) + δ= 2β + 2γSince the Euclidean Parallel Postulate holds, σ(△ABC) = 180, henceβ + γ = 90Corollary 41.2 If the vertices of a triangle △ABC lie on a circle and ABis a diameter of the circle then ∠ACB is a right angle. aa Euclid book 3 proposition 31: An angle inscribed in a semicircle is a right angle.Proof. This follows immediately from the previous theorem because themidpoint of AB is the center of the circle, hence AO = BO = CO.219