2-metric Axioms for Plane Euclidean Geometry

✍ Scribed by Patrick Cassens; B. A. Cassens

• Publisher: John Wiley and Sons
• Year: 1971
• Tongue: English
• Weight: 679 KB
• Volume: 51
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19710510103

✦ Synopsis

If B(a, b, c), then -B ( a , c, b).

A 8. If points a, b and c are collinear and distinct, then at least one of B ( a , b, c), B ( b , c, a) or B(c, a, b) is true. In view of T 2.2, T 2.3, and T 2.4 exactly one of B ( a , b, c), B ( b , c, a ) , B ( c , a , b) is true. T2.5. If B ( a , b, d ) and B ( b , c, d ) , then B ( a , b, c). Proof. Assume B(a, b, d ) and B ( b , c, d ) . Then a $. b + d f a, a b d = 0 and for each e E S, (1) Also, b f c + d += b, b c d = 0 and for each e E S, (2) e b c + e c d = e b d . From symmetry of the A function both b d a and b d c are zero. By A 3, since b + d and b d a = b d c = O , d a c = b a c = O . So a , b and c a r e collinear. Since a + b and b + c, to show a, b and c are pairwise distinct, assume a = c. Then B(u, b , d ) and B ( b , a , d ) which contradicts T 2.3. Thus exactly one of B ( a , b, c ) , B ( b , c, a ) and B(c, a , b ) is true. Assume B ( b , c, a). Therefore, B(n, c, b ) and for each e in S e n b f e b d = e a d . (3) From (I), (2), (3) and symmetry it follows that (4) By the simplex inequality ( 5 ) e a d g e a c f e d c f a d c .