triangle from 3 parallel lines

[shahar]

Can I make a proof that from 3 parallel lines you can't have 3 sizes (as part of parallel lines)that create a triangle of any kind?

 

[lev888]

Can I make a proof that from 3 parallel lines you can't have 3 sizes (as part of parallel lines)that create a triangle of any kind?

Sorry, don't understand the question. Lines? Not segments? Size as part of parallel lines? What?

 

[shahar]

which ways I can show logically that I can't construct a triangle from 3 parallel lines?

 

[Dr.Peterson]

How about having no vertices because the lines don't intersect?

 

[shahar]

So, How can I show it by formal proof?

 

[Dr.Peterson]

Make an attempt, and show us. This is not that hard.

First, state your proposition clearly. What does it mean to "construct a triangle from 3 parallel lines"?

Then state the definition of a triangle. What definition are you using?

Then find an axiom involving parallel lines and their intersections. What axioms (postulates) are you using?

That is where it has to start. Since I don't know what foundation your proof has to stand on, I can't start it for you. (There are several different sets of postulates or axioms that your textbook might use.)

 

[shahar]

So that:
1. A triangle is 3 sides with 3 interaction points. (by definition).
2. Let sign it by m = 3. (m - number of vertex).
3 parallel line are have 0 or infinite (both case are when the lines united).
4, Let sign it by a = 0 and b = ∞.
So, m != a and m! = b, so we have contradiction Q.E.D.

Question:

If I try to make x! =xn and y! = yn which x is every x of the point that different from x and y is every y that different so I to do it
I get the graph equation (by intuitive way):
y = x.
How I get it?.
Is my intuitive thinking is right.
so, every like that I draw with (xn, y) point is paralllel to the line y = x
when xn is different from x
and:
(x, yn) parallel to the line x = y
when yn is different from y

 

[Dr.Peterson]

Let me rephrase this as I understand it:

So that:
1. A triangle is 3 sides with 3 intersection points. (by definition).
2. Let's call it m = 3. (m - number of vertex).
3 parallel lines have 0 or infinite intersections (second case is when the lines coincide).
4, Let call it a = 0 and b = ∞.
So, m ≠ a and m ≠ b, so we have contradiction Q.E.D.

Question:

If I try to make x ≠ x_n and y ≠ y_n which x is every x of a point that is different from x and y is every y that is different so I try to do it
I get the graph equation (by intuitive way):
y = x.
How do I get it?
Is my intuitive thinking right?
so, every like that I draw with (x_n, y) point is parallel to the line y = x
when x_n is different from x
and:
(x, y_n) parallel to the line x = y
when y_n is different from y

But I can't make sense of what you are saying!

Are you trying to do a proof using coordinates? That might be valid; but you haven't answered my questions about the context. Any proof must be based on some axioms, but I don't know what you are assuming to be true.

What subject are you studying, into which this proof should fit?

As for the supposed proof in the first part, that is not what a proof looks like. There is no need to assign a variable to a number; proofs do not have to be symbolic.

I would say what I think you are saying in this way instead:

Proposition: A triangle can not be formed by segments of three parallel lines.

A triangle by definition is a figure composed of three vertices joined by three line segments. If two of the segments are on parallel lines, then those segments can't intersect, which violates the requirement that any two segments of a triangle must meet at a vertex.

(You seem to be allowing a line to be considered parallel to itself; I wouldn't have taken it that way.)

 

[shahar]

O.K. Now, I understand.
I have a question:
-If I need to use a sign or variable, it is only when I need to draw a picture, and include this variable sign in the picture, Right?
[*I mean that using of sign is only made in a picture that I draw and add to the solution]

The second question isn't right way of thinking.
I try to draw parallel line to point(x,y) and that point is of continuous function.
So, the only function that do it:
y = x
because all the parallel lines in the x-axis
and
the all parallel in the y-axis, create on the contunious function:
y = x
But now I think that I am wrong.

 

[Dr.Peterson]

O.K. Now, I understand.
I have a question:
-If I need to use a sign or variable, it is only when I need to draw a picture, and include this variable sign in the picture, Right?
[*I mean that using of sign is only made in a picture that I draw and add to the solution]

Symbols are needed for a variety of reasons, not just in geometry where they might refer to points or lines. They can be used for variables in algebraic work, representing unknown numbers or variables, for example. But there is (usually) no value in "naming" a number that already has a name, such as defining x=2.

The second question isn't right way of thinking.
I try to draw parallel line to point(x,y) and that point is of continuous function.
So, the only function that do it:
y = x
because all the parallel lines in the x-axis
and
the all parallel in the y-axis, create on the contunious function:
y = x
But now I think that I am wrong.

I have no idea what you are trying to prove here. But "parallel line to point(x,y)" is meaningless (you have to say what other line it is parallel to), as is "that point is of continuous function" (a point is not a function, and if you mean the point is on the graph of a continuous function, you have to identify that function, e.g. "continuous function y = f(x)").

Again, what do you mean by "parallel lines in the x-axis"? If you mean, "parallel to the x-axis", no -- y=x is not parallel to y=0.

 

[Steven G]

I haven't read most of the posts. I do not see why you can't make a triangle from three parallel lines. On each line pick one point. As long as the points are not co-linear, the three points can be the vertices of the triangle.

 

[pka]

Can I make a proof that from 3 parallel lines you can't have 3 sizes (as part of parallel lines)that create a triangle of any kind?

To shahar, You have really beaten a dead horse in this thread. Is your English proficiency up to the task of understand the answers given?
Do you understand the meaning of the statement \(A,~B.~\&~C\) are three non-colinear points?
The statement that two points determine a unique line is a universally true statement in geometry.
Therefore the points \(A,~B.~\&~C\) determine three distinct lines, any two of which have a point in common.
Thus no two of those lines are parallel. Moreover, those three points determine a triangle: \(\Delta ABC\).
Conversely the sides of the triangle \(\Delta XYZ\): \(\overleftrightarrow {XY},\,\overleftrightarrow {XZ}\,~\&~\overleftrightarrow {ZY}\) any two of which have a point in common. Thus no two can be parallel.
Does that no answer your question by putting a final nail in the coffin of your dead horse?

 

[shahar]

So, let try to revive the question (or horse). one, two, tree... electric shock.
If I change the triangle in the to plane.
Can I your way I can said that if I can construct (???) a triangle from 3 parallel lines, can I do it to a plane?
Can I create a plane by 3 parallel line?
I think that you can't break a whole geometric create (line) to a small create (like point on the line), (In physiology in called Gestalt).
There is a Israeli man that call it Organic Math. His name is Moshe Klein.
Is this thing here too? or no?

 

[shahar]

Counterexample that show that in the way you suggest there is one solution we need to thing about is: "When the all the 3 points of parallel line is on the same line".
the arrows show the points of that case!
(One question: "Which of you hear the news on Conway, who invented "Game of life" and etc by Coronavirus. R.I.P.)

 

[Deleted member 4993]

(One question: "Which of you hear the news on Conway, who invented "Game of life" and etc by Coronavirus. R.I.P.)

I did not - thanks for sharing....

 

[shahar]

Now, when I read it more than tree times, I figure out that you answer my question. My reading skills in English have a been problematic.
Terms like non-linear points force me not only to tranaslate it, to know what you meant before and how it connect to the next part of texts.
But every time I post here a message I learn new thing.
I can say that this term, not found and taught in schools and books in Israel, I found it only in one Enyclopedy.
So, It attack my contentraction skills in reading the post.
Good Day.

I can say that in Israel there is the Holocuast Day, Memorial Day and after these sad days, the Independance Day (of
Israel).