Plane Geometry - Bruce E. Shapiro

Section 19Angle BisectorsDefinition 19.1 (Angle Bisector) Let A, B, C be distinct non-collinearpoints. The ray −→ AD is the angle bisector of ∠BAC if D is in the interiorof ∠BAC and ∠BAD = ∠DACFigure 19.1:−→ AD is the angle bisector of ∠BAC if α = β.Theorem 19.2 (Angle Bisector Existence and Uniqueness) IfA, B, C are distinct and non-collinear then there exists a unique angle bisectorfor ∠BAC.Proof. To prove existence, define the measures of the various angles as illustratedin figure 19.1. We observe that by the angle construction postulatethere exists a unique ray −→ AD such thatβ = ∠BAD = 1 2 ∠BAC95