1. ## Betweeness of points

Hi, I am just working on random problems to study for my midterm, but then I came across this one I don't know how to do. Plz help me. Thankkk youuu sooo much.

1) Suppose that A, B, C are collinear points in R^3 whose coordinates are given by (a1, a2, a3), (b1, b2, b3), and (c1, c2, c3) respectively. Prove that A*B*C holds if we have a1 < b1 < c1, a2 = b2 = c2, and a3 > b3 > c3.

Many thanks.

Jen

2. Originally Posted by jenjen
Prove that A*B*C holds if we have a1 < b1 < c1, a2 = b2 = c2, and a3 > b3 > c3.
Jen
What does A*B*C mean?

3. Ohh, A * B * C means: B is between A and C

4. Originally Posted by jenjen

1) Suppose that A, B, C are collinear points in R^3 whose coordinates are given by (a1, a2, a3), (b1, b2, b3), and (c1, c2, c3) respectively. Prove that A*B*C holds if we have a1 < b1 < c1, a2 = b2 = c2, and a3 > b3 > c3.
I do not know how to prove something like this. But one of the undefined terms in Hilbert's geometry is "betweenness". However, it may be definied for numbers in $\mathbb{R}^n$ The only thing I can say is that since it is one a line and the x coordinates are increasing means the largest ones contain the smaller one.

5. Well, TPH this question may be beyond your experience.
In any metric space ‘betweeness” is a well-defined concept.
A*B*C means that d(A,B)+d(B,C)=d(A,C).

Hilbert’s axioms are for a synthetic geometry.
This question is not about synthetic geometry.
In traditional mathematics, $R^3$ is not synthetic geometry.

6. Originally Posted by Plato
Well, TPH this question may be beyond your experience.
In any metric space ‘betweeness” is a well-defined concept.
A*B*C means that d(A,B)+d(B,C)=d(A,C).
I actually understand you and a nice definition indeed.

The metric here is the regular distance metric?
$d(\bold{u},\bold{v})=\sqrt{\sum_{k=1}^n (a_k-b_k)^2}$

7. Ohh I am soo sorry, my definition wasn't so clear.