OME 1971 Problem 2

In a certain geometry we operate with two types of elements, points and lines, related to each other by the following axioms:I. Given two points $A$ and $B$ , there is a unique line $(AB)$ that passes through both.**II.**There are at least two points on a line. There are three points not situated on a straight line.III. When a point $B$ is located between $A$ and $C$ , then $B$ is also between $C$ and $A$ . ( $A, B, C$ are three different points on a line.)IV. Given two points $A$ and $C$ , there exists at least one point $B$ on the line $(AC)$ of the form that C is between $A$ and $B$ .V. Among three points located on the same straight line, one at most is between the other two.VI. If $A, B, C$ are three points not lying on the same line and a is a line that does not contain any of the three, when the line passes through a point on segment [AB] , then it goes through one of the $[BC]$ , or it goes through one of the [AC] . (We designate by [AB] the set of points that lie between $A$ and $B$ .)

From the previous axioms, prove the following propositions: Theorem 1. Between points A and C there is at least one point $B$ . Theorem 2. Among three points located on a line, one is always between the two others.