Plane Geometry - Bruce E. Shapiro

38 SECTION 9. BIRKHOFF/MACLANE AXIOMSBirkhoff’s AxiomsUndefined Terms: point, line, distance, angle.1. Axiom of Line Measure. The points A, B, .. of any line l can be put into (1, 1)correspondence with the real numbers x so that |x B − x A | = d(A, B) for all pointsA, B. Here d(A, B) denotes the distance between the points A and B. In otherwords, you are allowed to use a rule to measure the length of a line.2. The point-line postulate. One and only one straight line l contains two given pointsP, Q (P ≠ Q).3. The Axiom of angle measure. The half lines l, m, .. through any point O can be putinto (1,1) correspondence with teh real numbers a mod 2π so that, if A ≠ 0 andB ≠ O are points of l and m respectively, the difference a m − a l mod 2π is ∠A0B.4. The Postulate of triangle similarity. If in two triangles, △ABC and △A ′ B ′ C ′ ,of for some constant k > 0, d(A ′ , B ′ ) = kd(A, B), d(A ′ , C ′ ) = kd(A, C), andalso ∠B ′ A ′ C ′ = ±∠BAC then also d(B ′ , C ′ ) = kd(B, C), ∠C ′ B ′ A ′ = ±∠CBA,∠A ′ C ′ B ′ = ±∠ACB.1. There are at least two points.MacLane’s Axioms on Distance2. If A and B are points, d(AB) is a nonnegative number (that gives the distancebetween the points).3. For points A and B, d(AB) = 0 if and only if A = B.4. If A and B are points then d(AB) = d(BA).MacLane’s Axioms on Lines1. A l ine is a set of points containing more than one points.2. Through two distinct points there is one and only one l ine.3. Three distinct points on a line if and only if one of them is between the other two.4. On each ray from a point O and to each positive real number b there is a point Bwith d(OB) = bMacLane’s Axioms on Angles1. If r and s are rays from the same point, then ∠rs is a real number (mod 360).2. If r, s, t are three rays from the same point, the ∠rs + ∠st = ∠rt.3. If r is a ray from O and c is a real number, then there is a ray s from O such that∠rs = c ◦ .« CC BY-NC-ND 3.0. Revised: 18 Nov 2012