Why are isosceles triangles called that — or called anything? Why is their class given a name? Why did they find their way into the Elements and every single elementary geometry text and course ever since? Did no one ever ask himself, "What use is this, or why is it interesting?"?
Here are some facts about isosceles triangles whihc you might think would serve as valid answers to the above question, and I will attempt to show that they do not:
- A triangle has two equal sides iff it has two equal angles. But that's of interest only because we're already looking at the one class (triangles with two equal sides) or the other (those with two equal angles). And, in any event, the statement of the theorem is not more interesting than its generalization, that the larger a side in a triangle, the greater the angle opposite it.
- Various facts about the isosceles right triangle. Fine, I'll grant that the isosceles right triangle is interesting. But that's insufficient reason to give the much broader class of isosceles triangles a name.
- Any triangle can be partitioned into $n$ isosceles triangles $\forall n>4$ — and various other recent results. Very nice, but isosceles triangles are, of course, in Euclid, so these don't really answer the question.