# Geometry - Help with theorem proof please

Lee33
Geometry -- Help with theorem proof please

## Homework Statement

Let ##A,B,C,D## be points. If ##\vec{AB} = \vec{CD}## then ##A=C##.

None

## The Attempt at a Solution

This question was a theorem in my book that wasn't proved. I am wondering how to prove it?

It is saying that the vertex ##A## must equal ##C## if the ray ##\vec{AB} = \vec{CD}##.

The definition I have for ray is:

##\vec{AB} = \vec{AB} \cup \{ C \in P \ | \ A-B-C\}.## Where ##A-B-C## means ##B## is between ##A## and ##C##. And ##P## is the set of points.

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Homework Helper
Hi Lee33! (your definition doesn't look quite correct)

Suppose A ≠ C

A is in ##\vec{CD}##, so … ? Lee33
tiny-tim - Can you elaborate a bit more please?

Homework Helper
Hi Lee33! Apply the definition you were given …
The definition I have for ray is:

##\vec{AB} = \vec{AB} \cup \{ C \in P \ | \ A-B-C\}.## Where ##A-B-C## means ##B## is between ##A## and ##C##. And ##P## is the set of points.

Suppose A ≠ C

C is in ##\vec{AB}##, so what can you say about A B and C ? Lee33
If C is in ##\vec{AB}## and ##C\ne A## then B is between A and C?

Mentor
The definition I have for ray is:

##\vec{AB} = \vec{AB} \cup \{ C \in P \ | \ A-B-C\}.## Where ##A-B-C## means ##B## is between ##A## and ##C##. And ##P## is the set of points.
This definition makes no sense to me. First off, why would ##\vec{AB}## be equal to itself union some other thing (unless the other thing happened to be the empty set).

Second, how do you interpret ##\{ C \in P \ | \ A-B-C\}##? Does | have its usual meaning of "such that" or am I missing something? An explanation, in words, would be helpful.

Third, where are these points? Are they on a line or are they in the plane?

Fourth, how do you get that A - B - C means that B is between A and C?

Lee33
Sorry, I will elaborate.

First question: If A and B are distinct points in a metric geometry then the line segment from A to B is the set ##\vec{AB}=\{C \in P \ | \ A-C-B \ or \ C = A \ or \ C = B\}##.

If A and B are distinct points in a metric geometry then the ray from A toward B is the set ##\vec{AB}=\vec{AB}\cup \{C\in P \ | \ A-B-C\}.##

Second question: Yes, it means such that. Let P be the set of points in a metric geometry, and let C be a point in P such that B is between A and C.

Third question: They are on a line.

Fourth: That is just a notation for convenience. ##A-B-C## just means B is between A and C.

I will add the definition of between-ness: B is between A and C if the distance ##d(A,B)+d(B,C) = d(A,C)##.

Homework Helper
Hi Lee33! (just got up :zzz:)
If C is in ##\vec{AB}## and ##C\ne A## then B is between A and C?

nooo, C is (strictly) between A and B ok, and if A is in ##\vec{CD}##, then … ? Lee33
If A is in ##\vec{CD}## then A is between C and D.

Homework Helper
If A is in ##\vec{CD}## then A is between C and D.

yes (strictly between) ok, now you have two statements, and you should be able to prove a contradiction (thereby showing that "A ≠ C" was false) (drawing yourself a diagram might help)

Lee33
Alright thanks for the help! I will use your hints.

Question. Do I use both statements in my proof? That is, suppose ##A\ne C## and A is in ##\vec{CD}## then A is bewteen C and D. Also, I will use if ##A\ne C## and C is in ##\vec{AB}## then C is between A and B?

Homework Helper
Question. Do I use both statements in my proof? That is, suppose ##A\ne C## and A is in ##\vec{CD}## then A is bewteen C and D. Also, I will use if ##A\ne C## and C is in ##\vec{AB}## then C is between A and B?

yes gopher_p
You are using the same notation for line segment and ray, and it's confusing the bejeesus out of the people who are trying to help you.

Might I suggest ##\overline{AB}## for the segment and ##\overrightarrow{AB}## for the ray so that ##\overrightarrow{AB}=\overline{AB}\cup \{C\in P \ | \ A-B-C\}.##

Homework Helper
… it's confusing the bejeesus out of the people who are trying to help you.

it's not confusing me gopher_p
it's not confusing me Are you sure? Like, really sure? Because when Lee asked

If C is in ##\vec{AB}## and ##C\ne A## then B is between A and C?

you replied

nooo, C is (strictly) between A and B which is generally false regardless of which of Lee's two definitions of ##\vec{AB}## you're using. :tongue:

Homework Helper
… which is generally false …

well, Lee33 didn't contradict me, sooo i assume i got it right! Lee33
gopher_p - Sorry about that, you're right!

tiny-tim - If ##A\ne C## then ##C\in \vec{AB}## thus ##A-C-B## but where will the point ##D## be?

Homework Helper
but you haven't used …
If A is in ##\vec{CD}## then A is between C and D.

Lee33
So my proof should go like this:

Suppose ##A\ne C##, now since ##\vec{AB}=\vec{CD}## then ##A\in \vec{CD}## and ##C\in \vec{AB}##. Thus ##C-A-D## and ##A-C-B## which is a contradiction?

if I'm understanding the terminology correctly, you can't have both ##A-C## and ##C-A## unless C = A • 