Talk:Line (Euclidean geometry)/Archive 1

Rather cryptic

"The following demonstrates a line:

Given a line AC 
Point B is on AC
ABC is the same line AC"

— rather cryptic; what could it mean? Boris Tsirelson 19:39, 27 March 2010 (UTC)

Lead

Boris, could you provide a short lead? If I do it then I will probably not be allowed to approve it. Somewhere the long form "straight line" should also be mentioned. --Peter Schmitt 23:07, 11 May 2010 (UTC)

I shall try. Boris Tsirelson 05:53, 12 May 2010 (UTC)

Betweenness

The two conditions assume "three different points", thus the "at least" could be omitted. --Peter Schmitt 23:18, 11 May 2010 (UTC)

Sorry, I do not understand which "at least" do you mean.

"If three different points belong to the given set then at least one of them lies between the others" — if you mean this, well, I can delete "at least"; it will be a bit less formal but still clear.

"If one of three different points lies between the others, and at least two of the three points belong to the given set, then the third point also belongs to the given set" — well, really it is not mine "at least", I took it from WP (if I remember correctly). I can remove "at least", but probably the fear is that someone may say: it is contradictory, it cannot be that the number of these points on the given set is both two and three. No, after thinking more I see another fear related to Remark 3. If "at least" will be removed maybe we should say "some two of the three points" or maybe "any two of the three points"? The fear is that the reader can interpret the phrase as the weaker condition of Remark 3. Boris Tsirelson 06:27, 12 May 2010 (UTC)

First condition: For (pairwise) distinct points it will always be one point (or none). Thus the "less formal" version is equivalent and easier to read for non-mathematicians.

In the second case, too, I cannot see how "at least" may help. There is the danger that it is confused with convexity, but what has the "at least" to do with it?

I would vote for the least formal but still correct version, but I do not insist on omitting the "at least" (or any other change).

Perhaps "Among three distinct points of the given set there is always one that lies between the two others."?

Afterthought: the "distinct/different" is not needed either (even though the "at least" is then less formal).

And "If one of three distinct given points lies between the two others, and if any two of these three points belong to the given set, then the third point also belongs to the given set."

Or "If one of three given points lies between the two others, and if at least two of these three points belong to the given set, then the third point also belongs to the given set."

(Replacing the assumption "distinct" by the condition "at least"). --Peter Schmitt 10:55, 12 May 2010 (UTC)

On another matter: What about mentioning that the "betweenness" (as defined here) is related to the notion of "shortest path"? --Peter Schmitt 10:55, 12 May 2010 (UTC)

Yes, I agree, and did. I only disagree with the last version: "If one of three given points lies between the two others, and if at least two of these three points belong to the given set, then the third point also belongs to the given set." What if A=B, C is different, and A, B are on the line? Well, it seems, the matter is settled anyway. Boris Tsirelson 14:42, 12 May 2010 (UTC)

Definition via right angles

A link to Pythagorean theorem seems to be appropriate. --Peter Schmitt 23:41, 11 May 2010 (UTC)

Done. Boris Tsirelson 06:11, 12 May 2010 (UTC)