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I've looked in a math book that an isosceles triangle has at least two congruent sides. I also know that the words "at least" mean this symbol: $\ge$, which means "is greater than or equal to" or "is no less than." This got me thinking that equilateral triangles can also be isosceles triangles, but is that true?

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  • $\begingroup$ How many congruent sides does an equilateral triangle have? Is it at least 2? $\endgroup$
    – JRN
    Oct 26, 2014 at 13:53
  • $\begingroup$ In fact, it must be! $\endgroup$ Oct 26, 2014 at 13:56
  • $\begingroup$ Have you tried to find an equilateral triangle which is not isosceles? Thinking about it that way maybe help you to understand how the definitions work. $\endgroup$ Oct 26, 2014 at 13:58
  • $\begingroup$ Joel Reyes Noche, I know that an equilateral triangle has three sides, which is greater than or equal to two, so an equilateral triangle must be an isosceles triangle, too! $\endgroup$
    – Mathster
    Oct 26, 2014 at 14:01
  • $\begingroup$ Yes, that is correct. But you have to be careful. Some people (notably primary school teachers) define isosceles triangles to be those that have exactly two congruent sides. If that is the case, then for them, equilateral triangles are not isosceles. $\endgroup$
    – JRN
    Oct 26, 2014 at 14:02

4 Answers 4

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NB: I am presenting this answer as a frame challenge. The primary motivation behind this answer is to make more permanent some of the comments left in response to the question and other answers, as well as to incorporate some ideas from a now deleted answer.


The Importance of Definitions

Mathematics is a human endeavor. The words we use to describe mathematical ideas are a human invention, hence it is important to recognize that different humans might use the same word to describe different ideas, or different words to describe the same idea. When one is trying to understand a mathematical idea presented by another, it is important to understand the presenter's definitions. From the definitions, further deductions may be made.

For example, in the question above, we have the definition:

Definition: An isosceles triangle is a triangle with at least two congruent sides.

An equilateral triangle has three congruent sides, and three is "at least" two. Therefore, per this definition, every equilateral triangle must be isosceles.

However, there are authors who give a different definition of isosceles triangles. Joel Reyes Noche notes that many primary school instructors define an isosceles triangle to be one with exactly two congruent sides. Indeed, this is the definition given by Euclid himself!:

  1. Further, 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. [Euclid's Elements, as translated by Thomas Heath]

Per this definition, no isosceles triangle is equilateral, and no equilateral triangle is isosceles.

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An equilateral triangle is one with three equal sides. An isosceles triangle is one with two equal sides.

Therefore, every equilateral triangle is isosceles, but not every isosceles triangle is equilateral.

So far, so book. However, according to Wikipedia the definition of an isosceles triangle sometimes specifies that it must have two and only two equal sides. Under that (uncommon) definition, an equilateral triangle, having three equal sides, would of course not be isosceles.

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  • $\begingroup$ Well, because an equilateral triangle has three equal sides, two equal sides can be taken to say that an equilateral triangle is always an isosceles triangle. $\endgroup$
    – Mathster
    Oct 26, 2014 at 14:00
  • $\begingroup$ Exactly, but please note the "uncommon definition" paragraph in my updated answer. $\endgroup$
    – user139000
    Oct 26, 2014 at 14:03
  • $\begingroup$ I disagree with that page of Wikipedia because, as mentioned in the question I asked, I found that in a math book, an isosceles triangle has at least two congruent sides. $\endgroup$
    – Mathster
    Oct 26, 2014 at 14:03
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    $\begingroup$ Yes, under the definition you gave, every equilateral triangle is isosceles. $\endgroup$
    – user139000
    Oct 26, 2014 at 14:05
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    $\begingroup$ @Mathster I'm certain. Euclid defines isosceles triangles in Definition 20 of Book I of Elements. And I quote, "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." $\endgroup$
    – David H
    Oct 26, 2014 at 14:58
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From Shanghai (China), I have to say that equilateral triangle is a special case of isosceles triangle. Period.

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All equilateral triangles are isosceles because they have two congruent sides and two congruent angles

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