I've just spent the past few days wrestling with inclusive definitions for quadrilaterals, reading many, many past posts on this forum (thankyou John Conway, Walter Whiteley and others!). Having accepted the inclusive approach for quadrilaterals, I'm now turning back to triangles and looking at these anew.
My initial thoughts are that, looking at sides, then an isosceles triangle is a triangle with *at least* 2 congruent sides. Following this, an equilateral is an isosceles triangle with *exactly* 3 congruent sides. Of course, symmetry could be used as the feature to focus on, but the result is essentially the same, isn't it?
But where does a scalene triangle fit? Does it sit at the top of a hierarchy, with the isosceles and equilateral as special cases? I'm thinking yes, as whatever theorems that apply to scalene triangles should apply to the others. But how would a scalene triangle be defined by itself so that it still allows these special cases?
Also, because triangles can be defined in terms of their largest angle (acute, obtuse or right), would there be any hierarchy in terms of angles?