# Geometry - Help with theorem proof please (1 Viewer)

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#### Lee33

Geometry -- Help with theorem proof please

1. The problem statement, all variables and given/known data

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

2. Relevant equations

None

3. 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 … ? #### Lee33

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 ? #### Lee33

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?

#### 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 $d(A,B)+d(B,C) = d(A,C)$.

#### 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 … ? #### Lee33

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)

#### 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?

#### 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 #### 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 $\overrightarrow{AB}=\overline{AB}\cup \{C\in P \ | \ A-B-C\}.$

#### tiny-tim

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:

#### tiny-tim

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?

#### tiny-tim

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?

#### tiny-tim

Homework Helper
yes!!

if i'm understanding the terminology correctly, you can't have both $A-C$ and $C-A$ unless C = A • 1 person

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