6. There exist three distinct points A, B, C not in any of the orders ABC, BCA, CAB.
| Fig. 73. |
7. If three distinct points A, B, C (fig. 73) do not lie on the same line, and D and E are two distinct points in the orders BCD and CEA, then a point F exists in the order AFB, and such that D, E, F are collinear.
Definition.—If A, B, C are three non-collinear points, the plane ABC is the class of points which lie on any one of the lines joining any two of the points belonging to the boundary of the triangle ABC, the boundary being formed by the segments BC, CA and AB. The interior of the triangle ABC is formed by the points in segments such as PQ, where P and Q are points respectively on two of the segments BC, CA, AB.
8. There exists a plane ABC, which does not contain all the points.
Definition.—If A, B, C, D are four non-coplanar points, the space ABCD is the class of points which lie on any of the lines containing two points on the surface of the tetrahedron ABCD, the surface being formed by the interiors of the triangles ABC, BCD, DCA, DAB.
9. There exists a space ABCD which contains all the points.
10. The Dedekind property holds for the order of the points on any straight line.
It follows from axioms 1-9 that the points on any straight line are arranged in an open serial order. Also all the ordinary theorems respecting a point dividing a straight line into two parts, a straight line dividing a plane into two parts, and a plane dividing space into two parts, follow.
Again, in any plane α consider a line l and a point A (fig. 74).