# Geometry - Help with theorem proof please

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.

Last edited:

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

Suppose A ≠ C

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

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

Mark44
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?

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)##.

tiny-tim
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 … ? If A is in ##\vec{CD}## then A is between C and D.

tiny-tim
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)

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?

tiny-tim
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 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\}.##

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

it's not confusing me 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:

tiny-tim
Homework Helper
… which is generally false …

well, Lee33 didn't contradict me, sooo i assume i got it right! 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?

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

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?

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