# Proof that three parallel lines don't be cutted by a transversal in Klein model

How do you prove that three parallel lines don't be cutted by a transversal?

By definition parallel are Chords that meet on the boundary circle are limiting parallel lines. Then I built three paralles lines:

Then I can draw a line that cut them.

Otherwise, if lines a and b are parallel to d, but they have a common point. Then:

• Certain triples of parallel lines can be cut by a transversal. I assume you want to show that there exists a triple that cannot be cut in this way. Do you know how to tell whether two chords meet at a right angle in the Klein model? – Cheerful Parsnip Mar 16 '16 at 18:12
• If @GrumpyParsnip's interpretation of your question is correct: Consider a Euclidean triangle with vertices on (or outside) the boundary circle. – Blue Mar 16 '16 at 18:35
• Hello GrumpyParsnip, I know how draw perpendiculars in Klein Model and @Blue you are right, a transversal can be cut one parallel or two parallels or three parallels... In euclidean plane alway a transversal cut these triples – Salvattore Mar 17 '16 at 13:17
• @Salvattore: you need to put an "@" in front of my name for me to get a message. I just happened to come back to this question and saw you commented. – Cheerful Parsnip Mar 19 '16 at 2:50

To long for a comment therefore added as answer:

Your question is unclear, I think you do not really understand the question and that you need to analyse the question more, what (very) exactly is what is given and what is asked for?

I suggest you carefully write out the givens give the elements names and write the situation out in basic statements.

Also analyse what is asked for (Is asked for a proof/an example/ a counterexample that given the givens, something is always/ is sometimes/ is sometimes not/ is never the case?

Maybe after you have analysed the question this way you do not need any help anymore at all. Sadly sometimes geometry is more textual analyse than mathematics.

I am wondering: Is the question you need to answer not in english and because you do not really understand the question you are not able to translate it?

Maybe the question is: (and to give examples how the question look after analysing)

• given 2 non intersecting lines $a$ and $b$ and a transversal $c$ cutting both $a$ and $b$. Show in the Klein model of hyperbolic geometry that a line $d$ exist that is not intersected by $a$, $b$ and $c$.

or

• given 2 intersecting lines $a$ and $b$ and a line $c$ cutting both $a$ and $b$. Show in the Klein model of hyperbolic geometry that a line $d$ exist that is not intersected by $a$ , $b$ and $c$ .

(do you see the subtile differences?)

Also notice that the question on first reading is likely to put you on the wrong foot. (because the question is to give an example that given a situation something is not always the case, this is where the textual analyse and sometimes some psychology could be helpful)

to end with more general help:

• If you have to prove that two lines do not always intersect use short chords for them.

• if you have to prove that two lines can intersect use diameters for them.

Hopes this helps

More help:

Was thinking a bit more:

• Lines on in the Klein model (and also in the Poincare disk model) can be identified by their endpoints.
• If one line intersects another the endpoints seperate eachother (going round the boundary circle it looks like (an end of line a-an end of line b-an end of line a-an end of line b)
• if one line does not intersects another the endpoints do not seperate eachother (going round the boundary circle it looks like (a-a-b -b , a-b-b-a , they are the same)
• this way the whole question just becomes a question to put the endpoints in the right order on the boundary circle,
• draw the connecting lines between the endpoints
• and you are done.
• The question is: – Salvattore Mar 17 '16 at 12:55
• The question is: "Given 3 parallel lines a, b ,d. Show in the Klein model of hyperbolic geometry thath a line e exist that is not intersect the three parallels a, b, c in the same time". Thanks for your comment, I understand more clearly... I need proof that e maybe cut one parallel or two parallels, not always cut a, b and d. – Salvattore Mar 17 '16 at 13:31
• So the question is: "Given 3 parallel (non intersecting) lines $a$, $b$ and $c$. Show in the Klein model of hyperbolic geometry that there exist a line $e$ that does not intersect the lines a, b, c at the same time". Keep all chords as short as reasonable and you cannot go wrong. You can even make lines $a$, $b$ and $c$ horoparallel (meeting at the boundary circle) , notice that a line that intesects both outside lines will intersect the inside line. but you can construct a line that intersects the middle line but not the outside lines (make a diameter of the middle line) , good luck – Willemien Mar 17 '16 at 17:26