Is an equilateral triangle an isosceles triangle?

[Harry_the_cat]

This is an ongoing disagreement between two friendly mathematicians. I'd like to hear others' thoughts on the matter.

I've always believed an isosceles triangle is defined as a triangle with at least two equal sides. Therefore an equilateral triangle is an isosceles triangle.

I am having difficulty convincing my friend who believes an isosceles triangle is defined as a triangle with (exactly) two sides and as a result thinks that an equilateral triangle is NOT an isosceles triangle.

I've found lots of internet sites that agree with my definition, but can't rely on many of these sites.

Is there a definitive definition (if that's a phrase) of an isosceles triangle?

 

[pka]

This is an ongoing disagreement between two friendly mat.hematicians. I'd like to hear others' thoughts on the matter. I've always believed an isosceles triangle is defined as a triangle with at least two equal sides. Therefore an equilateral triangle is an isosceles triangle. I am having difficulty convincing my friend who believes an isosceles triangle is defined as a triangle with (exactly) two sides and as a result thinks that an equilateral triangle is NOT an isosceles triangle.

Is a parallelogram also a trapezoid?
Is a square also a rhombus?
Is an equilateral triangle also an isosceles triangle?
Well it depends upon whom one asks.

E H Moore alone with his PhD student R L Moore(no relation) are the founders of American axiomatic geometry. Moore the elder was invited to give a lecture at Princeton. He began is talk with "Let a be a point and b be a point. The German emigre mathematician Leteschetz yelled from the back of the lecture hall "but why don't you just 'let a & b be points'". Moore replied, "because a could be b" , at which point Leteschetz stormed out of the room.

So you are stuck with having to read an author's set of definitions. Sorry.

 

[mmm4444bot]

I've always believed an isosceles triangle is defined as a triangle with at least two equal sides. Therefore an equilateral triangle is an isosceles triangle.

I am having difficulty convincing my friend who believes an isosceles triangle is defined as a triangle with (exactly) two sides and as a result thinks that an equilateral triangle is NOT an isosceles triangle.

Wikipedia acknowledges both views:

In geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having two and only 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.

Euclid defined an isosceles triangle as one having exactly two equal sides, but modern treatments prefer to define them as having at least two equal sides, making equilateral triangles (with three equal sides) a special case of isosceles triangles. In the equilateral triangle case, since all sides are equal, any side can be called the base, if needed, and the term leg is not generally used.

​

I note the phrase "modern treatments".

One beauty of mathematics is its plasticity. Math evolves.

Is there a definitive definition (if that's a phrase) of an isosceles triangle?

Apparently not, but that shouldn't be an issue, when authors make clear which definition's in play.

Like, sometimes 0 is a negative number; sometimes it's not.

Sometimes LOG denotes base 10; sometimes it doesn't.

Sometimes 0^0 is undefined; sometimes it's not.

Sometimes 0 is a Natural number; sometimes it's not.

... :cool:

 

[mmm4444bot]

... having to read an author's set of definitions. Sorry.

That's not something we need to apologize for. :cool: