Plane Geometry - Bruce E. Shapiro

72 SECTION 14. BETWEENNESSorf(A) > f(C) > f(B).If f(A) > f(C) > f(B) then−AC = f(C) − f(A) > f(B) − f(A) = −ABso AB > AC which is a contradiction.If f(A) < f(C) < f(B) thenAC = f(C) − f(A) < f(B) − f(A) = ABwhich is also a contradiction. Hence A ∗ C ∗ B is also not possible. All thatis left is A ∗ B ∗ C.Definition 14.28 The point M is the midpoint of the segment AB if A ∗M ∗ B and AM = MB.Theorem 14.29 (Existence and Uniqueness of Midpoints) If A andB are distinct points then there exists a unique point M that is the midpointof AB.Proof. To prove existence, let f be a coordinate function for the line ←→ AB,and definex =f(A) + f(B)2Since f is onto, there exists some point M ∈ ←→ AB such that f(M) = x.Hence2f(M) = f(A) + f(B)orf(M) − f(B) = f(A) − f(M)Thus AM = MB. To see that A ∗ M ∗ B, leta = min(f(A), f(B))b = max(f(A), f(B))Since A and B are distinct then a ≠ b and we havewith a < b. Hencex = a + b2x < 2b2 = b« CC BY-NC-ND 3.0. Revised: 18 Nov 2012