Comments on: Does isosceles subsume equilateral?

Question: Does isosceles subsume equilateral?

i_am_joe's_spleen — Thu, 03 Mar 2005 11:11:56 -0800

Are equilateral triangles also isosceles?

My daughter was given homework in which she had to identify isosceles triangles. We decided that given the definition "has two equal sides", an equilateral triangle must also be isosceles. Her maths teacher disagrees. It's been a long time since I did plane geometry, but I'm sure that the formal definition was not "two sides only"
I can't find authoritative, unambiguous references on Google. I'd love an online reference, but a dead-trees one would be good too.

I'm not going to buy a fight over geometry with a hapless primary school teacher but this is driving me nuts.

By: modofo

modofo — Thu, 03 Mar 2005 11:18:48 -0800

I don't know how authoritative Wolfram Research is, but this is definitely unambiguous (and agrees with you).

By: Leon

Leon — Thu, 03 Mar 2005 11:19:21 -0800

An equilateral triangle is therefore a special case of an isosceles triangle

By: muhonnin

muhonnin — Thu, 03 Mar 2005 11:20:48 -0800

Mathworld says that an equilateral triangle is a special case of an isosceles triangle. So your daughter has a pretty good case.

By: true

true — Thu, 03 Mar 2005 11:24:27 -0800

Unless its a Canadian triangle, then the teacher may be right. (Scroll down)

By: stopgap

stopgap — Thu, 03 Mar 2005 11:24:36 -0800

Intuitively, I'd say you're right and the teacher is wrong.

This page at PlanetMath says:

Triangles can be classified according to the number of their equal sides. So, a triangle with 3 equal sides is called equilateral, triangles with 2 equal sides are isosceles and finally a triangle with no equal sides is called scalene. Notice that an equilateral triangle is also isosceles, but there are isosceles triangles that are not equilateral.

This Google search is showing a few other quotes that imply the same thing (e.g. "Any equilateral triangle
is also isosceles & scalene...", "Note that an equilateral triangle is also isosceles and acute.")

By: Leon

Leon — Thu, 03 Mar 2005 11:25:24 -0800

modofo by a nose...

By: nobody

nobody — Thu, 03 Mar 2005 11:28:33 -0800

Primary school curricula often involve over-classification (for example, simile vs. metaphor). If pressed, the textbook writers or department heads would probably call it pedogogical necessity.

The interesting question now is: now that you and your daughter know that you are correct and that your daughter's teacher is, strictly speaking, wrong, what will/should you do?

By: PinkStainlessTail

PinkStainlessTail — Thu, 03 Mar 2005 11:28:48 -0800

I'm sure that the formal definition was not "two sides only"

Unfortunately, it sounds to me like the teacher is trying to enforce this definition in order to keep things "simple," rather than taking the time to explain to the kids how one figure can be two things at once. Do they have a similar issue with calling squares rectangles, even though squares are a specific type thereof?

By: profwhat

profwhat — Thu, 03 Mar 2005 11:30:44 -0800

The granddaddy of all Western geometry is Euclid, and he said that:

Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

Under this definition, an isosceles triangle has "two of its sides alone equal," so the teacher is right. But note the commentary: "As defined by Euclid, an equilateral triangle is not to be considered as an isosceles triangle, but in modern terminology, it is usually the case that equilateral triangles are included among the isosceles triangles, that is, it is only required that at least two sides be equal in order for a triangle to be isosceles. Generally speaking, modern definitions are inclusive whereas Euclid's definitions are usually exclusive."

By: andrew cooke

andrew cooke — Thu, 03 Mar 2005 11:34:55 -0800

your daughter is right
but she should learn how to get good marks. which means knowing the right answer, recognising that teachers are often not as smart as she is, identifying what they "really want", and giving them it.

By: andrew cooke

andrew cooke — Thu, 03 Mar 2005 11:38:38 -0800

(the reason modern definitions are typically inclusive is that it makes more practical sense - if someone decides, tomorrow, that triangles with two sides equal and an included angle of 90 degress are to be called "george triangles" then if you use an exclusive definition, it has changed (since george triangles are george triangles, and not isocoles triangles). in contrast, with inclusive definitions, you don't have to keep re-writing things when you define a new (sub-)class of objects.)

By: Eamon

Eamon — Thu, 03 Mar 2005 11:46:23 -0800

This is even a better lesson for your daughter than "one figure can be two things at once": two disagreeing people can be correct at once. And definitions change. And sometimes, you need to shut up and nod your head even though you're right and they're wrong.

By: sciurus

sciurus — Thu, 03 Mar 2005 12:01:20 -0800

So is a square not a rectangle?

By: PinkStainlessTail

PinkStainlessTail — Thu, 03 Mar 2005 12:04:04 -0800

"Don't let school interfere with your education."

By: Coffeemate

Coffeemate — Thu, 03 Mar 2005 12:27:48 -0800

"A square is always a rectangle, but a rectangle is not always a square" is what I was always taught.

By: andrew cooke

andrew cooke — Thu, 03 Mar 2005 12:43:29 -0800

(and also, is saves time and effort. if you prove something about isocoles triangles, it will apply to equilateral triangles too. so by making the definition inclusive, you don't have to keep saying "and equilateral too, of course".)

By: LionIndex

LionIndex — Thu, 03 Mar 2005 13:19:03 -0800

You can flesh out the square/rectangle thing further by adding in that a square is also a rhombus, and that all those things are basically just parallelograms.

Also: circle vs. ellipse

By: vacapinta

vacapinta — Thu, 03 Mar 2005 13:39:18 -0800

and also, is saves time and effort. if you prove something about isocoles triangles, it will apply to equilateral triangles too.

You mean like: Exactly two of the three interior angles of an isosceles triangle will be equal. :D

By: andrew cooke

andrew cooke — Thu, 03 Mar 2005 14:14:34 -0800

mathematician.
and i mean that in a bad way.

By: denishowe

denishowe — Thu, 03 Mar 2005 18:28:32 -0800

Is a bicycle a kind of tricycle?

By: Eideteker

Eideteker — Thu, 03 Mar 2005 20:19:01 -0800

Recall the scene from Little Man Tate where the teacher has numbers 1-10 written on the board and asks, "Which of these are divisible by two?" The child answers: "All of them," because the teacher did not stipulate they be evenly divisible.

It's not about wrong or right, but rather how you back up your reasoning. That is what children should be taught in school. Remember, "show your work." I always got in trouble because I never showed my reasoning for problems (I called them "trivial"), which is a good thing, because it made me learn to back up my arguments when we got to calculus and things got more nebulous and less intuitive.

By: Aknaton

Aknaton — Thu, 03 Mar 2005 20:31:52 -0800

(IAAM.) The basic principle to follow in mathematical definitions is that you should set them up to make theorems easier to state and remember. If we followed Euclid's ugly terminology, theorems that could nicely end "...then T is isosceles" would have to instead end "...then T is isosceles or equilateral."

Why is 1 not considered prime? Because we like the theorem that says "evey positive integer is the product of a unique list of primes, up to reordering" more than one that would say "a unique list of primes other than 1". Feh.

Instruct your daughter to look for moments when the teacher says "and therefore this triangle is isosceles" to say "you haven't checked that the third side is different from the other two -- are you ready yet to rethink your ill-considered opinion on the proper definition of isosceles?"

By: epimorph

epimorph — Thu, 03 Mar 2005 21:32:58 -0800

In math, definitions are a matter of opinion. There is really no definitive definition of any concept in math, even with things like natural numbers. What professional mathematicians do is use whatever definitions they think are best, but they always make it clear what definitions are being used. Similarly, in a math class the teacher/professor gets to set the rules and dictate the definitions, and the students are stuck with those definitions whether or not they like them. The only way the teacher can be at fault here is if he or she did not explicitly state which definition was being used or if he or she provided a definition that is ambiguous.

By: Kwantsar

Kwantsar — Thu, 03 Mar 2005 23:00:29 -0800

everything reduces to NAND

By: i_am_joe's_spleen

i_am_joe's_spleen — Thu, 03 Mar 2005 23:33:30 -0800

thank you everyone. believe me, i know that there are more lessons here than maths ones.