Talk:Isosceles trapezoid

"intersect at equal positions"

What does "intersect at equal positions" mean? — Timwi 01:32, 8 Mar 2004 (UTC)

A little late, but I think what is meant by this is that the two top sections of the diagonals are congruent, and the two bottom sections of the diagonals are also congruent. All four diagonal sections are not congruent, as can be seen by drawing a tall, skinny isosceles trapezoid. Loggie 01:29, 4 May 2005 (UTC)

A little long to be a stub

Are you sure this is a stub? It seems a little long to be a stub yet too short to be a full-fledged article. --Mannyram24 13:36, 6 April 2006 (UTC)

the problem of parallelograms

The article suggests that a paralleologram is not an isosceles trapezoid. Why? A parallelogram has two parallel opposing sides, and the other two sides are equal to each other. —Preceding unsigned comment added by Honghaier (talk • contribs) 17:15, 13 October 2007 (UTC)

In essence the idea is to distinguish between two types of quadrilaterals, namely Trapezoids and Parallelograms. If parallelograms were isosceles trapezoids, then they would have properties like diagonals of the same length, or congruent base angles. Most parallelograms do not have the diagonals of the same length, or base angles congruent unless they are rectangles. In this case, isosceles trapezoids and rectangles do have the same properties. The bottom line: when you write a property of an isosceles trapezoid you are not writing a property of a parallelogram, but a rectangle; so it makes no sense to call parallelograms isosceles trapezoids since the former do not have the same properties of the latter. —Preceding unsigned comment added by 72.178.193.150 (talk) 04:10, 12 August 2009 (UTC)

The distinguishing property is the line of symmetry (parallelograms in general don't have one). It remains true that some parallelograms (namely rectangles) are isosceles trapezoids (or trapeziums outside the USA), but a general parallelogram is not. The intersection of the two sets is the set of rectangles. Dbfirs 11:45, 12 August 2009 (UTC)

What makes a trapezoid a trapezoid is not the line of symmetry. Look for a book of geometry and you will see that the line of symmetry is a property, not part of the definition (hence, the definition of the isosceles trapezoid is wrong in this article). What makes a trapezoid such is the fact there are two parallel sides. Some people think that parallelograms should be trapezoids because they satisfy that part of the definition, and sometimes (for some purposes) they both have similar properties (e.g. they are both convex quadrilaterals), but the distinction is necessary when you want to prove properties like base congruent angles, etc. However, what should be clear is that there is no general agreement about if parallelograms are trapezoids. The main point is why would you want parallelograms to be trapezoids. This has to do with theory (e.g. properties that they have). I still fail to see what is the advantage of making parallelograms trapezoids, when parallelograms do not have base congruent angles, so it is a bit of a contradiction (to put it lightly) to prove a theorem like "isosceles trapezoids have congruent base angles" and then say "well, parallelograms are isosceles trapezoids, but unless they are rectangles they do not satisfy the above property about angles". It is better to make a clear distinction and say "trapezoids have only to parallel sides", and this avoids all the problems that you could have with exceptions to some of the properties that trapezoids have but parallelograms do not. —Preceding unsigned comment added by 72.178.193.150 (talk) 03:21, 15 August 2009 (UTC)

You seem to be presenting two conflicting views. The definition in the article is designed specifically to preclude the admission of the general parallelogram from the set of isosceles trapezoids (trapeziums), but to include the rectangle. Why do you think that this definition is wrong? Dbfirs 08:28, 24 August 2009 (UTC)

There are two schools about what a trapezoid is. It all depends on your needs. From the perspective of a mathematician (as opposed to a lay person), the need is dictated by the theory, not personal preferences, or tradition. In fact, theories have changed a lot through the years. Calculus did not have the form it has until the end of the 19th century, so you can imagine how long it took to put it in the terms that it exists today. Even up to this day, there has not been a traditional book on Calculus that has been studied for centuries, so all corrections that have had to be made to the theory have prevailed over the original attempts.

In constrast, Geometry has had a main book that has determined the tradition over centuries. That tradition was changed at the end of the 19th century and beginning of the 20th century when Hilbert et al. rewrote the theory in Geometry. The new theory in Geometry is written to include the old theory, in other words, all the current knowledge of geometry includes Euclid's books.

The problem that one has with trapezoids is if there are only two parallel sides, or should there be two pairs of parallel sides. Again, think about needs and theory. What are your needs? If you are trying to classify shapes (as in different shapes look essentially different), or if you are trying to write formulas for lengths, areas, etc. and these formulas look too different between quadrilaterals with only two parallel sides, than with parallelograms, and if the former type of quadrilaterals appear too much in your theory, then it is natural to give them a name that distinguish them from the other type of quadrilaterals.

What this all means is that you must judge your need before you create a definition. I can define many things, but only those that are relevant are the ones to be considered. Consider the following chain of definitions and theorems

Definition: A trapezoid is a quadrilateral that has only one pair of parallel sides. The non parallel sides of a trapezoid are called legs.

Definition: A parallelogram is a quadrilateral that has both pair of opposite sides parallel.

Definition: An isosceles trapezoid is a trapezoid, whose legs have the same length.

It is clear from this definition that parallelograms are not isosceles trapezoids.

Ok, now that definitions have been laid out, we can prove theorems. Here are some theorems

Theorem: in an isosceles trapezoid, the diagonals have the same length.

If parallelograms were to be considered isosceles trapezoids, that theorem would be false. In some place in the proof of that theorem one needs to use that isosceles trapezoids are not parallelograms. I will leave that to you to figure out where that is done.

One could reply back and say, wait, Rectangles do satisfy that, they are parallelograms and satisfy that, so why can't I call rectangles "isosceles trapezoids". Again, the point is the need. Do we need to include rectangles in the theory of isosceles trapezoids? Are rectangles trapezoids? not according to the definition given above.

Ok, so let me look at the flip side. Assume that we give the following definition

Definition: a trapezoid is a quadrilateral that has at least two parallel sides.

In that definition, a parallelogram is a trapezoid. Now you want to define isosceles trapezoid, so how do you that? Well, you want to preserve the theorem that in an isosceles trapezoid the diagonals have the same length. You realize that you can not do this by using the length of the other sides, since in most parallelograms, the diagonals have different lengths, so you are forced to make a different type of distinction between quadrilaterals, and that is that they have a specific line of symmetry. What has been done from the perspective of theory, is that you need to create a number of definitions to explain what symmetry means. This includes the idea of perpendicularity, so you need new definitions and criteria in order to distinguish an isosceles trapezoid from within the class of trapezoids. It takes a bigger effort to define an isosceles trapezoid according to this theory, than is necessary. I showed before that it is not necessary. All you need is the concept of parallelism, not of perpendicularity or symmetry.

This should make the first theory preferable over the second.

The only advantage that I have ever seen of using the second definition is that with the second definition you can have the following theorem

Theorem: All trapezoids are convex quadrilaterals.

In the first theory, you have to prove separate theorems for trapezoids and parallelograms. I do not consider that to be a big disadvantage (or one could prove the theorem that "if a quadrilateral has two parallel sides, then it is convex", which includes the trapezoid (in the first sense) and the parallelogram.

I hope this makes sense. —Preceding unsigned comment added by 72.178.193.150 (talk) 15:51, 30 August 2009 (UTC)

Yes, it makes perfect sense. It's just that all of the mathematicians I've ever met use inclusive definitions (your second definition) for quadrilaterals. Dbfirs 11:27, 23 May 2010 (UTC)

I don't understand this assertion: In that definition, a parallelogram is a trapezoid. Now you want to define isosceles trapezoid, so how do you that? Well, you want to preserve the theorem that in an isosceles trapezoid the diagonals have the same length. You realize that you can not do this by using the length of the other sides, since in most parallelograms, the diagonals have different lengths.... [emphasis added]

A parallelogram with unequal diagonals (i.e. a parallelogram other than a rectangle) is not an isosceles trapezoid, so what's the problem? —Tamfang (talk) 18:42, 24 May 2010 (UTC)

Paragraph moved from article

I've moved the following paragraph from the article: An isosceles trapezoid is also a quadrilateral such that the diagonals divide each other into segments of different length that are congruent among each other. In the picture above, this means that if $AE=DE$, $BE=CE$ and $AE\not =CE$, then the quadrilateral is an isosceles trapezoid.
Whilst I understand what the anon editor was trying to say, I have two issues:

1. the phrase congruent among each other is vague, and so is not good mathematics. The property is also menioned earlier in the article.
2. The condition $AE\not =CE$, is not necessary according to our definition.

What does anyone else think? Dbfirs 08:22, 24 August 2009 (UTC)

Well, if you do not like a phrase, improve it; do not remove it. More to the point, the condition $AE\not =CE$ is given because trapezoids are not parallelograms. If $AE=CE$ (and $BE=DE$), then the quadrilateral is a parallelogram, not a trapezoid. This brings me to the main point. The definition of isosceles trapezoid in this article is not the correct one. That definition is a property. More specifically, we can agree to study quadrilaterals that have a line of symmetry as stated in this article, but those are different type of quadrilaterals, not isosceles trapezoids. In other words, if you want to use such definition, do not use it to define isosceles trapezoids. To give a complete answer, an isosceles trapezoid is a trapezoid (as in "a quadrilateral that has only two parallel sides) where the non parallel sides have the same length. We can talk more about it, if you like. —Preceding unsigned comment added by 72.178.193.150 (talk) 14:55, 30 August 2009 (UTC)

It is considered improper, on talk pages, to edit the comments of others to reverse the meaning.Dbfirs 08:26, 31 August 2009 (UTC)

I've done as you suggest (with the help of Tamfang). Dbfirs 08:28, 31 August 2009 (UTC)

If AE=CE (and other defining conditions are met) it's a rectangle, which is an isosceles trapezoid and a parallelogram. —Tamfang (talk) 15:50, 30 August 2009 (UTC)

See my response above as to why to make isosceles trapezoids not be parallelograms. —Preceding unsigned comment added by 72.178.193.150 (talk) 15:53, 30 August 2009 (UTC)

Um, obviously we don't want to define trapezoids as parallelograms, but you haven't stated a reason to define trapezoids in such a way as to exclude rectangles. —Tamfang (talk) 16:33, 30 August 2009 (UTC)

Our article on quadrilaterals specifically shows rectangles as a subset of isosceles trapeziums (trapezoids in the USA). I see no reason to exclude rectangles as a subset since rectangles inherit all properties of the isosceles trapezoid. The current Wikipedia definition of the isosceles trapezoid seems to have been carefully constructed (five years ago) to precisely match the article on quadrilaterals. Dbfirs 08:08, 31 August 2009 (UTC)

The main idea is that trapezoid is the quadrilateral that you obtain when you divide a triangle in two by a parallel line to one of the sides (one is a similar triangle to the original triangle, the other a quadrilateral which is a trapezoid). Point is, the trapezoid was not supposed to be a parallelogram. Parallelograms, instead have a different idea behind them. There is no such thing as a trapezoid that is a parallelogram (or viceversa). Of course, you can define things to your convenience so that rectangles be trapezoids, but what have you gained from it? The theory has not been improved, has been made more confusing. Particularly, for those poor souls that do not really understand what is really going. It seems that I can't win just because you refuse to hear. I wish you had a better reason than what you are giving. Really. I really wished I could learn something other than sometimes you loose because the person in power does not listen. To summarize, the definition is wrong. It leads to more problems than it solves. I am sorry you were not better advised at the time the definition was written. Enough.72.178.193.150 (talk) 02:09, 5 October 2009 (UTC)

I see a difference between "If you do this you get a trapezoid" and "A trapezoid is what you get if (and only if) you do this". The definition of a geometrical figure ought to be its properties, not its construction (there's more than one way to construct anything). — Please note that a rectangle can be made from a degenerate isosceles triangle with a vertex at infinity. —Tamfang (talk) 15:16, 19 October 2009 (UTC)

The inclusive definition given in the article (and in the quadrilateral article) is the one that is standard in the UK, but I think American mathematicians avoid the exclusive definition for their trapezoids (our trapeziums). It is probably not important enough to argue about. Dbfirs 11:27, 23 May 2010 (UTC)

rectangles

I'd support a careful discussion in the trapezoid section about rectangles. As it stands we should acknowledge that rectangles are in a gray area (without decreeing the correct answer). For that matter a triangle might be considered a trapezoid with one side of length zero although I doubt there would be much suport for that.--Gentlemath (talk) 21:15, 7 December 2009 (UTC)

Unless all sources agree on "the correct answer" we should not provide one ourselves, rather we should describe both sides of any disagreement that might exist. See WP:NPOV and WP:OR. —David Eppstein (talk) 00:24, 8 December 2009 (UTC)

So you agree with me, thanks! Or maybe not. The article appears reverted although I don't see it in the history. In fact when I look at the article text itself it says:

An isosceles trapezoid (isosceles trapezium in British English) is a quadrilateral with a line of symmetry bisecting one pair of opposite sides,

making it automatically a trapezoid. Two opposite sides (bases) are parallel, the two other sides (legs) are of equal length.

but when I consider editing it it says:

An isosceles trapezoid (isosceles trapezium in British English) is a quadrilateral with a line of symmetry bisecting one pair of opposite sides,

making it automatically a trapezoid. Many sources, but not all, would qualify all this with the exception: "excluding rectangles."

Anyway, is the intention something like: An isosceles trapezoid is a (convex) quadrilateral with exactly one line of symmetry, that line bisecting a pair of opposite sides? The way it was written one day ago seemed to include rectangles, which are the one class of parallelograms which some would admit as trapezoids. I didn't take a stand one way or another, merely said: some would say 'but not rectangles'. I'll look for a source with the minority opinion . For now I think I will open an edit and then save without doing anything just to see what happens.--Gentlemath (talk) 03:12, 8 December 2009 (UTC)

OK,now the text does include my change. The book The classification of quadrilaterals: a study of definition By Zalman Usiskin, Jennifer Griffin, David Witonsky, Edwin Willmore is viewable on google books. A summary at http://www.mathcurriculumcenter.org/QuadSummary.pdf shows that it discusses the question are rectangles isosceles trapezoids. Some sources (such as the influential text by Moise in foundations of geometry) use an inclusive definition which considers every parallelogram as a trapezoid. More conservatively one could observe that if convex quadrilateral ABCD has sides AB and DC parallel and sides AD and BC congruent then either r angles A and B are congruent (isosceles trapezoid) or else angles A and C are congruent (arbitrary rectangle). The former case can also be characterized as the one in which the figure can be inscribed in a circle. But in the special case of a rectangle both apply. Ultimately it is a matter of convention. In the case it is not clear which convention is to be preferred. --Gentlemath (talk) 03:43, 8 December 2009 (UTC)

For what it's worth, my own preference is to be inclusive. It keeps the definitions simpler that way. But I agree that we should let the literature rather than our own preferences guide us here. —David Eppstein (talk) 04:04, 8 December 2009 (UTC)

I agree that the inclusion of rectangles is just a matter of convention (and the wording of the definition), but I was under the impression that the vast majority of sources and almost all mathematicians used inclusive definitions. Is this not the case? Dbfirs 08:13, 9 December 2009 (UTC)

I don't actually know. My suspicion is that the distribution of answers would be different if you asked research mathematicians vs if you asked the writers of high school geometry textbooks. —David Eppstein (talk) 08:12, 10 December 2009 (UTC)

One could allow 3 conventions 1) a parallelogram is never a trapezoid 2) rectangles are trapezoids but not other paralleograms are 3) all parallelograms are trapezoids I am fairly sure that almost every current US textbook uses convention 1). So that should maybe be noted to avoid confusing students (hence I would revert the latest change). I personally think 2) makes sense since the main theorem becomes (parallel bases + equal base angles)<=>(parallel bases + equal legs)<=>isosceles trapezoid. Also then every quadrilateral in the trapezoid rule is a trapezoid! Since b-a=0 in this case, some of the formulas (with (b-a)^2 in a denominator) need a fix (cancel out the (b-a)^2 from the numerator). (gentlemath at a public computer) —Preceding unsigned comment added by 131.91.228.172 (talk) 23:08, 9 December 2009 (UTC)

Those (b-a)^2 formulas are in the trapezoid article. —Preceding unsigned comment added by 131.91.228.172 (talk) 23:11, 9 December 2009 (UTC)

I think 3) is common in the UK (as at the end of our article on quadrilaterals), but I agree that we need to report differing definitions. I'm surprised that US textbooks use option 1) - is this just so that they don't have to make an exception to a formula? Dbfirs 08:08, 10 December 2009 (UTC)

I see it differently than what is presented here. The overall set would be Trapezoids (having a pair of opposite sides parallel) to separate it from the remaining quadrilaterals. Then that set is divided into two disjoint sets, Parallelograms and Non-parallelogram Trapezoids (traditional trapezoids). The Non-parallelogram Trapezoid set has the proper subset Isosceles Trapezoid. A rectangle would not be an Isosceles Trapezoid maintaining the oblique nature of "isosceles". However, if you prefer a crossover, it would be no different than saying a square is a kite. Regarding your question on the definition, the answer is most likely inertia. Not enough agreement for texts to change (speculation). I have no idea regarding the situation for advanced geometry texts. JackOL31 (talk) 23:43, 12 December 2009 (UTC)

That's an interesting alternative veiwpoint that I hadn't thought of. I suppose that the trapezoid (or trapezium in the UK) is just not important enough to be consistently defined. Dbfirs 00:19, 13 December 2009 (UTC)

I really regret not being able to participate in this conversation when it originally happened! :) I have recently been doing a lot of research on the trapezoid definition struggle. Of course, like most other mathematicians, I think the inclusive definition is the "right" one, and that the obvious fix for defining an isosceles trapezoid is to use "base angles congruent." The fact that it's not equivalent to "legs congruent" does not seem like a major loss. Rschwieb (talk) 16:19, 21 November 2012 (UTC)

Bad definition

In Euclidean geometry, an isosceles trapezoid (isosceles trapezium in British English) is a trapezoid where the legs have equal length.

Under this definition, a non-rectangular parallelogram would be an isosceles trapezoid. Georgia guy (talk) 18:38, 18 March 2013 (UTC)

Yes, see the comment above. It's OK if you define "legs" as non-parallel sides, but then the set of rectangles would not be a subset of the isosceles trapezoids. The recent change by David Eppstein sorts out the problem. Dbfirs 22:50, 18 March 2013 (UTC)

Parallelograms again

I gather from the previous discussions on this talk page that isosceles trapezoids must be defined to exclude non-rectangular parallelograms. Currently this article's lead says

In Euclidean geometry, an isosceles trapezoid (isosceles trapezium in British English) is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides.

Not sure, but it seems that this permits parallelograms. The article Symmetry (geometry) says

Under an isometric transformation, a geometric object is said to be symmetric if, after transformation, the object is indistinguishable from the object before the transformation, i.e., if the transformed object is congruent to the original.[5]

Doesn't either of the (bimedian) lines bisecting two opposite sides of a parallelogram cut it into congruent pieces? Then the parallelogram is symmetric, and each bimedian is a line of symmetry, so the article's current definition includes parallelograms.

Maybe this could be fixed by changing "line of symmetry" in our definition to "line of reflectional symmetry"? Loraof (talk) 17:34, 9 April 2016 (UTC)

A parallelogram that is neither a rectangle nor a rhombus has no lines of symmetry. A line of symmetry (in 2 dimensions) means reflection symmetry. Rotational symmetry is defined by a point, not a line, unless we're dealing with 3-dimensional figures. Georgia guy (talk) 17:43, 9 April 2016 (UTC)

Right, our redirect Line of symmetry reflects (ahem) what you say. Loraof (talk) 21:59, 9 April 2016 (UTC)

By the way, the definition of a symmetry transformation quoted above is wrong. The transformed object needs to be equal to the original, not just congruent to it. —David Eppstein (talk) 23:43, 9 April 2016 (UTC)

What's the difference between "equal to" and "congruent to"? Loraof (talk) 19:32, 11 April 2016 (UTC)

Congruence is precisely defined in Mathematics (see Congruence (geometry), though I expect you already know the definition). "Equal to" is not clearly defined for shapes because one needs to specify which properties are equal. I've removed the misleading "equal to" in the article, and also the misleading reference to congruence. Dbfirs 20:58, 11 April 2016 (UTC)

Citation 2 (Referenced under "Special Cases") is inappropriate.

The reference in question excludes rectangles only in that it uses the exclusive definition of trapezoid - rectangles are being excluded because the text does not consider them trapezoids at all (much less isosceles trapezoids). Since Trapezoid defaults to the inclusive definition, I'd assume we'd want to maintain that here. If we want to note that an exclusive definition of isosceles trapezoid excludes rectangles, then we should say that explicitly. It is unclear if that was the intended meaning here, or if this also refers to other definitions that exclude rectangles for whatever reason (in which case, we need citations of those).66.211.251.59 (talk) 19:35, 7 July 2022 (UTC)