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?
[Math] Can an equilateral triangle be an isosceles triangle, too
definitionterminologytriangles
Related Solutions
Let $ABC$ be a triangle with $\angle BCA=\angle ABC$. Then observe that $ABC\equiv ACB$ (the order of the vertices is important), because they satisfy $ASA$ criterion. Therefore, we have that $CA=AB$, i.e. $ABC$ is isosceles.
I don't believe you'll find debate on the in/exclusiveness of "scalene" the way you do with "isosceles" vs "equilateral" (or "rectangle" vs "square", or "trapezoid" vs "parallelogram", etc). The term seems to exist precisely to guarantee that there are no symmetries that can cause elements —say, an altitude, median, and/or angle bisector from a vertex— to coincide, so that constructions relying on distinct elements don't degenerate. At the moment, I can't think of a particularly-good example that "goes bad" for isosceles triangles, but I'll note that the Euler line is undefined for equilaterals, due to the coincidence of all the key "triangle centers" that would otherwise uniquely determine it.
Incidentally, according to the "Earliest Known Uses of Some of the Words of Mathematics" pages, "scalenum" excluded isosceles from way-back-when.
In Sir Henry Billingsley’s 1570 translation of Euclid’s Elements, scalenum is used as a noun: "Scalenum is a triangle, whose three sides are all unequall."
It might be interesting to check Elements to see to what degree the passages using the term needed the triangles in question to be asymmetric.
As for whether you should (ahem) "[infer] the solution must never work when $\triangle ABC$ is isosceles or equilateral" ... It's possible that symmetry will cause a problem's solution to fail completely (think Euler line), but it's also possible that an isosceles/equilateral configuration amounts to a "limiting form" of the property under discussion; in the latter case, such a form might well be considered valid from a suitable point of view, but the presenter may not (yet) want to burden the discussion with such nuances. So, never infer "never".
In, say, textbook exercises or contest problems, my sense of things is that the term "scalene" is often the author's assurance that the reader needn't fear gotcha degeneracies.
Best Answer
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:
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!:
Per this definition, no isosceles triangle is equilateral, and no equilateral triangle is isosceles.