Is an equilateral triangle an isosceles triangle?

Harry_the_cat

Harry_the_cat

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A square is a rectangle. Right! I think we all agree on that (although I have argued with students over that.)

Is an equilateral triangle an isosceles triangle?

An isosceles triangle is defined as having two equal length sides.

Does this mean "exactly two" or "at least two"?

In the first case, an equilateral triangle is not isosceles, but in the second case it is.

I've always believed it is, ie an equilateral triangle is a special sort of isosceles triangle, in the same way that a square is a special sort of rectangle.

It all comes down to the definition.

Comments?
 
Harry_the_cat said:
… It all comes down to the definition …
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I agree.

I also consider an equilateral triangle a special case of isosceles. (I don't use the word 'exactly', as shown below.)

From wiki: "Sometimes [an isosceles triangle] is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case."
 
Harry_the_cat said:
A square is a rectangle. Right! I think we all agree on that (although I have argued with students over that.) Is an equilateral triangle an isosceles triangle? An isosceles triangle is defined as having two equal length sides. Does this mean "exactly two" or "at least two"? In the first case, an equilateral triangle is not isosceles, but in the second case it is. I've always believed it is, ie an equilateral triangle is a special sort of isosceles triangle, in the same way that a square is a special sort of rectangle.It all comes down to the definition.
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If a triangle has three congruent sides it has two congruent sides.

If a person is asked if s/he has two siblings and the person says "yes", then it turns out they have four siblings are they truthful?
 
Harry_the_cat said:
A square is a rectangle. Right! I think we all agree on that (although I have argued with students over that.)

Is an equilateral triangle an isosceles triangle?

An isosceles triangle is defined as having two equal length sides.

Does this mean "exactly two" or "at least two"?

In the first case, an equilateral triangle is not isosceles, but in the second case it is.

I've always believed it is, ie an equilateral triangle is a special sort of isosceles triangle, in the same way that a square is a special sort of rectangle.

It all comes down to the definition.

Comments?
Click to expand...

In everyday life, we often assume "exactly", as in "I have two brothers" (not three), or "That's a square" (not a rectangle). But in science and math, hierarchical classifications are the norm, which requires inclusive rather than exclusive definitions. The definition of "isosceles" needs to include "equilateral" as a special case, so that when we state theorems about isosceles triangles, we don't have to always add that it applies to equilateral triangles as well.

But even a mathematician, when asked to draw an isosceles triangle, will probably not draw an equilateral one. That's a matter of context: in some situations, the natural interpretation demands the most specific meaning, rather than the inclusive meaning. And I think this is the cause of much of the confusion we see among students.

So the definition, in mathematical contexts, is "at least two congruent sides", but sometimes we tend to take it otherwise.

For a collection of interesting discussions of this sort of question (yours is covered in the last half), see my blog post here.
 
Dr.Peterson said:
But even a mathematician, when asked to draw an isosceles triangle, will probably not draw an equilateral one.
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And when most people (even mathematicians) are asked to draw a rectangle, they will draw one with horizontal and vertical lines, with the horizontal being the longer. I suppose it goes back to how we first see and learn about rectangles as a kid. Interesting.

And thanks for the link Dr P.
 
Are Harry_the_Cat and the chessire_cat member of same feline family??
 
Harry_the_cat said:
A square is a rectangle. Right! I think we all agree on that (although I have argued with students over that.)

Is an equilateral triangle an isosceles triangle?

An isosceles triangle is defined as having two equal length sides.

Does this mean "exactly two" or "at least two"?

In the first case, an equilateral triangle is not isosceles, but in the second case it is.

I've always believed it is, ie an equilateral triangle is a special sort of isosceles triangle, in the same way that a square is a special sort of rectangle.

It all comes down to the definition.

Comments?
Click to expand...
I am curious why you would think that a square is a rectangle but be troubled with whether or not an equilateral triangle is an isosceles
triangle. The definition of an isosceles triangle says that there are two equal sides. It does not say exactly 2. If it did, as Dr P already mentioned, then we would have to have the same theorem for equilateral triangles as we have for isosceles triangles and mathematicians are smart enough not to let that happen. Equilateral triangles are all isosceles triangles.
 
Jomo said:
I am curious why you would think that a square is a rectangle but be troubled with whether or not an equilateral triangle is an isosceles
triangle. The definition of an isosceles triangle says that there are two equal sides. It does not say exactly 2. If it did, as Dr P already mentioned, then we would have to have the same theorem for equilateral triangles as we have for isosceles triangles and mathematicians are smart enough not to let that happen. Equilateral triangles are all isosceles triangles.
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Its been an ongoing argument/discussion with a colleague. As I said, I believe equilateral triangles are isosceles, whereas my colleague believes they are not. We've never really been able to settle our argument as the definition seems a bit ambiguous.

It does not say exactly 2. If it did, as Dr P already mentioned, then we would have to have the same theorem for equilateral triangles as we have for isosceles triangles and mathematicians are smart enough not to let that happen.
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​

I'm not sure what you mean by the above?? All of the theorems for isosceles triangles hold true for equilateral triangles. They must, is equilateral triangles are a subset.
 
Harry_the_cat said:
Its been an ongoing argument/discussion with a colleague. As I said, I believe equilateral triangles are isosceles, whereas my colleague believes they are not. We've never really been able to settle our argument as the definition seems a bit ambiguous.
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I think mathematicians generally are so familiar with their own use of the language (e.g. that "having two equal sides" means "having at least two equal sides") that they forget to clarify it for the general public. This leads to definitions that are not ambiguous to the author, but are ambiguous to students, and therefore to teachers, who are trying to translate for the students.

One way to settle the argument would be to look at some theorems in your book that mention isosceles triangles in the conditions or in the conclusion, and observe either that the theorems would not be technically correct by the exclusive definition, or would be incomplete. This depends on what theorems (or exercises) the book gives, and how they are stated; a book might manage to work around all the problems, deliberately or accidentally, so that it would end up not mattering how they take the definitions. But I would expect them to slip up eventually, and say something that implies the proper definition (or that is wrong!).

Now, there are some similar situations where there is genuine ambiguity or disagreements among mathematicians. One such example, the trapezoid, is discussed in my followup blog post here.
 
Dr.Peterson said:
One way to settle the argument would be to look at some theorems in your book that mention isosceles triangles
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That was the problem and where our discussion started. We were actually writing a series of textbooks (for Years 8-10) and obviously trying to get our definitions spot on and consistent. Many textbooks, at that level anyway, have dodgy or watered-down definitions. We didn't want to be one of them.
 
Harry_the_cat said:
That was the problem and where our discussion started. We were actually writing a series of textbooks (for Years 8-10) and obviously trying to get our definitions spot on and consistent. Many textbooks, at that level anyway, have dodgy or watered-down definitions. We didn't want to be one of them.
Click to expand...

I fully agree! Especially at the elementary level, I've observed that many "definitions" aren't really definitions at all (or are missing entirely).

So you get to decide what theorems you want, and observe the effect of different definitions on how the theorems have to be stated (in order to be consistent with the definitions). That could be an interesting exercise.
 
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