Talk:Triangle

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External links modified[edit]

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External links modified[edit]

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any three non-collinear points determine a unique plane?[edit]

Article says: "In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space)." I believe the part about "unique plane" is incorrect, because any plane has an infinite number of triangles. The triangle from the three points is unique, because it is different (different vertices) from any other triangle. But the plane is not unique, because an infinite number of other triangles determine the same plane. If I misunderstand, someone please correct me. The correct wording, IMO, should be "In Euclidean geometry any three points, when non-collinear, determine a unique triangle." AAABBB222 (talk) 18:04, 14 August 2017 (UTC)

The statement in the article is quite correct, three non-collinear points do determine a unique plane. The fact that there are many triangles in a plane does not change this (consider this − two distinct points determine a unique line, but there are an infinite number of pairs of points on this line.) One of the easiest ways to see that this is true is to use analytic geometry (coordinates) and some linear algebra. --Bill Cherowitzo (talk) 18:38, 14 August 2017 (UTC)
To me it seems incorrect. In addition to what I already said, the article is about triangles, not planes. So why start the article with a statement about planes that is confusing at best? Furthermore, you can reverse the logic to say "In Euclidean geometry any triangle determines a unique set of three non-collinear points." However, trying to do the same for the plane statement fails: "In Euclidean geometry any plane determines a unique set of three non-collinear points." :( Finally, the word "determine" is a synonym for "define" here, and a triangle does not "define" a plane. The furthest I would go and keep the explanation clear is something like "A triangle lies on one, and only one, plane". AAABBB222 (talk) 21:38, 14 August 2017 (UTC)
I'm sorry, but your intuition is leading down the wrong path. A triangle does define a plane and your final statement is just another way to say that a triangle determines a unique plane. There are reasons to bring up planes in regards to triangles. To start with, it should be made clear that triangles are objects that live in planes. The same can not be said for sets of four or more points. Also, when you move away from Euclidean geometry, the statement about triangles living in planes is no longer always true, and this is brought up in the article. I have expanded that paragraph in the article somewhat, but I am not sure that this will satisfy you. --Bill Cherowitzo (talk) 23:18, 14 August 2017 (UTC)
I think we would be better served not making comments about my intuition, and sticking with the facts. A square also defines a plane. So does a pentagon, hexagon, etc. So I think it is not needed, and confusing, to introduce a triangle by suggesting it is somehow special because it "determines" a plane. The other polygon articles don't waste space on that, so neither should this one (Please do not go editing them to say they "determine a unique plane"). And there is more than three points to the concept of a plane, so I think it is incorrect to say that a triangle "defines a plane". It simply lies on a plane. And again reversing the logic of the statement does not work, as pointed out in previous comment, so the statement is confusing. A triangle is simply a three-sided polygon, period. Maybe I should have just edited the article. Anyone else? AAABBB222 (talk) 21:32, 23 August 2017 (UTC)
A square, pentagon, hexagon, etc. are all defined as plane figures, so yes they do determine a plane–the one they live in. However, in Euclidean 3-space, the four vertices of a quadrilateral do not determine a plane, nor do the correct number of vertices for any of the spacial polygons, except for the triangle. That is why it is important to bring this up. That three non-collinear points define a plane has a very clear mathematical meaning. See Plane (geometry)#Describing a plane through three points for the details. Your "reversing the logic" argument makes no logical sense. You are simply interchanging the words triangle and plane as if they are identical objects. A triangle determines a unique plane, but a plane contains many triangles (any one of which determines that plane). If all of geometry was limited to the plane, then your point might have some merit, but when higher dimensions are involved you must make allowances for the fact that not all planar concepts are still valid. --Bill Cherowitzo (talk) 22:14, 23 August 2017 (UTC)
I can only agree with Bill. Sometimes mathematical English makes use of constructions that do not sound very normal in non-mathematical English, but to get anywhere you have to know what is being meant. Double sharp (talk) 23:52, 23 August 2017 (UTC)
I agree that the article is accurate as it stands, but we will be happy to consider any improvements to avoid misunderstandings of what is stated there. Please discuss them here first. Dbfirs 08:08, 24 August 2017 (UTC)