Question

Prove the Converse of Proposition 3.3 by using Betweenness Axiom 1. The converse is Given B ? C ? D and A ? B ? D, then A ? B ? C and A ? C ? D. Please do not use "by mapping of letters"

Answer 1

(figure)

Given: and

To prove: and

Proof: (Method 1)

Now Betweenness Axiom 1: If , then A,B, and C are three distinct points all lying on the same line, and

According to the converse of proposition 3.3 If , then B, C and D are three distinct points.

We can infer from this that Pt C lies between B and D .....................................(1)

If , then A, B and D are three distinct points and Pt B lies between Pts A and D.

Now, as Pt C lies between B and D and Pt B lies between A and D. So it means that points B and C lies between A and D.

From statement 1, we could infer that it is Pt B which lie between A and C.

......................(2)

Now can also be ...............(3)

From 2 and 3, we can infer

Thus implies Pts A, B, C and D are four distinct points.

Method 2:

If then .......................(1) (Betweenness property)

and then ...................(2)

From1 and 2 ,

It means that Pts A, B , C and D are four distinct points such that

Thus proved