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Looking through the book "Euclidean and Non-Euclidean Geometries" by Marvin Jay Greenberg, there is the given problem:

Given two points A and B and a third point C between them. (Recall that "between" is an undefined term.) Can you think of any way to prove from the postulates [Euclid's I-V] that C lies on line $\overleftrightarrow{AB}$?

There are multiple ways I could go about trying to prove this, but I am wondering if it is even possible with the axioms that Euclid provided.

I was thinking that I could use Euclid's Postulate II to relate the "betweeness" to a line, but I wasn't sure if that was good approach.

Jeffrey Walraven
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  • Hint: what if it weren't on the line? How could you tell the difference between points that do and points that don't lie on the line. – fleablood Sep 05 '17 at 01:33
  • Thank you for responding so quickly! Are you suggesting that "C between them" may not be on the line AB since "between" is not clearly defined? – Jeffrey Walraven Sep 05 '17 at 01:38
  • I believe that Euclid "defines" a line as "that which lies evenly between its extremes" (or "its endpoints"). Perhaps you can build a satisfactory proof using this definition. – Jim H Sep 05 '17 at 01:44
  • If "between" means "is on the line" there is nothing to prove. If "between" means something that can be used to prove that "between" $\implies$ "on the line" then "between" has to have some definition. So I have no idea what this question even means. I took it to mean C is somewhere to the right of A, either on or off the line, and somewhere to the left B-- if it were on the line how could we prove it? If the qestion means something else then I don't know what. – fleablood Sep 05 '17 at 01:45
  • @fleablood Yes, I am also confused as to what it is asking. I think you are right that if it is on the line, then there is nothing to prove. But if it simply means that C is somewhere to the right of A and to the left of B, then it would be impossible to prove, since it could be off the line (like you suggested). – Jeffrey Walraven Sep 05 '17 at 01:48
  • @JimH I think that would be the most straightforward.Unfortunately, the book takes a very different approach than Euclid and states that a line is an undefined (primitive) term. – Jeffrey Walraven Sep 05 '17 at 01:52
  • I think what they are getting at is you can create the distances (radii) AC and BC and thus create a circle centered at A with radius AC + BC. If B is on this circle then C is on the line. Otherwise C is not. .... But I think I actually do not know what the book is asking and I'd need to read that chapter myself. – fleablood Sep 05 '17 at 01:56
  • That may be it. I will ask my professor tomorrow and see what he thinks. Thanks for all the help! – Jeffrey Walraven Sep 05 '17 at 02:03

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Here are the relevant statements from Greenberg's book.

EUCLID'S POSTULATE I. For every point $P$ and for every point $Q$ not equal to $P$ there exists a unique line $l$ that passes through $P$ and $Q$.

DEFINITION. Given two points $A$ and $B$. The segment $AB$ is the set whose members are the points $A$ and $B$ and all points that lie on the line $AB$ and are between $A$ and $B$.

EUCLID'S POSTULATE II. For every segment $AB$ and for every segment $CD$ there exists a unique point $E$ such that $B$ is between $A$ and $E$ and segment $CD$ is congruent to segment $BE$.

Given the above postulates, there is no way of proving that $C$ lies on line $AB$. Because if $C$ belongs to segment $AB$ then $C$ is between $A$ and $B$, but the converse needn't be true.

If this proof were possible, some of the betweenness axioms in Chapter 3 could be avoided. As stated on page 70: "In Exercises 6 and 7, Chapter 1, we saw that some assumptions about betweenness are needed".

Intelligenti pauca
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