Does the set of scalene triangles include the special cases of isosceles and equilateral triangles?

Question

A math question starts:

$\triangle$ABC is a scalene triangle ...

From this should I imply the solution must never work when $\triangle$ABC is isosceles or equilateral? That is to ask, does the set of scalene triangles include the special cases of isosceles and equilateral triangles?

Can an equilateral triangle be an isosceles triangle, too?

Answer 1

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.

Answer 2

One observation. If the set of scalene triangles included isosceles and equilateral triangles, then it would actually be the set of all triangles. So scalene triangle would just mean triangle and the term would be redundant.

Clearly this would be true whether or not isosceles triangles were defined as including equilateral ones.

A proof might consider scalene triangles in order to guarantee that it works for arbitrary side lengths—but this is different from defining all triangles as scalene.