MathDB

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.

Problem Statement