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.

  • $\begingroup$ 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. $\endgroup$ – fleablood Sep 5 '17 at 1:33
  • $\begingroup$ 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? $\endgroup$ – Jeffrey Walraven Sep 5 '17 at 1:38
  • $\begingroup$ 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. $\endgroup$ – Jim H Sep 5 '17 at 1:44
  • $\begingroup$ 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. $\endgroup$ – fleablood Sep 5 '17 at 1:45
  • $\begingroup$ @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). $\endgroup$ – Jeffrey Walraven Sep 5 '17 at 1:48

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".


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.