# Why are there isosceles triangles? [closed]

Why are isosceles triangles called that — or called anything? Why is their class given a name? Why did they find their way into the Elements and every single elementary geometry text and course ever since? Did no one ever ask himself, "What use is this, or why is it interesting?"?

Here are some facts about isosceles triangles whihc you might think would serve as valid answers to the above question, and I will attempt to show that they do not:

• A triangle has two equal sides iff it has two equal angles. But that's of interest only because we're already looking at the one class (triangles with two equal sides) or the other (those with two equal angles). And, in any event, the statement of the theorem is not more interesting than its generalization, that the larger a side in a triangle, the greater the angle opposite it.
• Various facts about the isosceles right triangle. Fine, I'll grant that the isosceles right triangle is interesting. But that's insufficient reason to give the much broader class of isosceles triangles a name.
• Any triangle can be partitioned into $n$ isosceles triangles $\forall n>4$ — and various other recent results. Very nice, but isosceles triangles are, of course, in Euclid, so these don't really answer the question.
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## closed as not constructive by Austin Mohr, Steven Stadnicki, Dilip Sarwate, Pedro Tamaroff, Asaf KaragilaMay 16 '12 at 12:07

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Equilateral triangles are regular polygons, so they stand out. At the other extreme are triangles with all three sides different. Isoceles triangles are exactly the ones "in between" the two extremes. "Isosceles" comes from the Greek isos (equal) and skelos (meaning "leg"), i.e., triangles with "equal legs". –  Arturo Magidin May 15 '12 at 20:17
"No one cares about pentagons with precisely three equal sides"... well, that's a rather sweeping statement, though to some extent true. On the other hand, people do care about quadrilaterals with all sides the same, sides partitioned into two sets of equal parts, or that have exactly three sides equal. As to what may or may not seem a weak reason, it all depends on your interest; if you happen to be interested in triangles, and you find that you can say a lot about equilaterals but not much about general scalene, being able to discuss the middle ground may not seem weak (cont) –  Arturo Magidin May 15 '12 at 20:22
(cont) Now, if your question is really "Why should I care about isosceles triangles?" then that's a different question altogether, and the answer is "No particular reason; if you don't want to care about them, then don't." People single out all sorts of "in between classes" all the time, because they find they can say interesting things about them. Trapezoids are singled out among general quadrilaterals, parallelograms among general polygons, prisms from among solids, convex polygons from among polygons, etc. –  Arturo Magidin May 15 '12 at 20:23
@msh210: "The special case is not especially interesting." I really do wish you would stop stating your personal lack of interest as if it were a universal truth. I do not doubt that many things I find interesting you would find boring, and I would expect you to realize the reciprocal truth. –  Arturo Magidin May 15 '12 at 20:35
I think you need to take care about a number of things. First, in Euclid's development of geometry from the axioms the facts about isosceles triangles are logically prior to facts about scalene triangles - you can't have those facts otherwise. Secondly Isosceles triangles have a symmetry which may be thought to have some elegance. Thirdly, triangles are rigid structures - it will not always be possible to fit an equilateral triangle into a structure, and an isosceles triangle might be an elegant alternative. –  Mark Bennet May 15 '12 at 21:54

If you're ever doing a geometrical construction involving two radii of a circle with their endpoints joined by a line, you'll probably need some facts about isosceles triangles.

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Nice; thanks; +1. –  msh210 May 15 '12 at 23:29

I believe that one of the reasons why isosceles triangles are discussed in the elements is because Euclid's construction of the regular pentagon hinges on the construction of an isosceles triangle with a nice (will edit with more specifics later) relationship between the length of its sides.

The Greeks were interested in constructing regular polygons. A regular $n$-gon is constructible if and only if $n$ factors into a power of $2$ and a product of distinct Fermat primes (Gauss). So the regular pentagon was largest 'building block' for constructing regular polygons that anyone discovered until Gauss. This is one of the reasons why it was significant, and as a result so were isosceles triangles.

On another note, reflecting a triangle over one of its sides is common in elementary geometry proofs. This yields an isosceles triangle. This happens often enough to warrant giving isosceles triangles a name to reference the particular properties they have that don't hold for triangles in general. (In other words, giving them a name makes many elementary geometry proofs shorter, even if the thing being proved isn't even about triangles.)

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Nice; thanks; +1. –  msh210 May 15 '12 at 23:29

Here are a couple of very practical reasons:

If you are an engineer and you have two long pieces of wood or metal which you want to secure a fixed distance from each other, you might just have to hand a number of standard pieces of the same length (I think about making a crane from a Meccano set). Then you are likely to create a structure which contains a number of isosceles triangles, with the exact geometry depending on the separation you want to achieve.

In an engineering design, the equal legs of an isosceles triangle will often be bearing the same load, and therefore need to be equally strong, stiff etc, so can be made of standard material.

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Equal sides occur commonly, e.g. in radii of a circle.

Using these equal sides to identify equal angles is useful and leads to various theorems about circles. In this sense, isosceles triangles are perhaps more useful than equilateral triangles. I am not an expert, but I could recommend that you read The Elements :) From my past experience, it's quite readable (visual), and I imagine that Euclid makes it pretty clear why he thinks these are interesting.

"No one cares about pentagons with precisely three equal sides" - perhaps, because these do not have any associated/convenient angle facts.

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I believe I did once study pentagons with three equal sides (likewise angles) as a non-trivial case I didn't have full intuition for. –  Mark Bennet May 15 '12 at 21:56