We need a new term that means "equilateral triangle." "Equilateral rectangle" gets its own term: "square." Why not "equilateral triangle"? "Square" is one of the first shapes you teach a little kid. But who asks a preschooler, "Can you find the equilateral triangle"? No one. Even though the triangle that you use to teach a kid what a triangle is is probably an equilateral triangle.
"Triangle" already is an unfairly technical term. It was obviously invented by somebody rather than sprouting out of common use, as "square" did. But I can live with "triangle," provided that the special, archetypal triangle gets a good name for itself.
Here are some possible names for "equilateral triangle":
Triad: Already means something, but maybe we could commandeer the word to mean equilateral triangle.
Trid: Sounds a lot like "trig," which could be a problem.
Hexadek: Tricky way to say that all the angles are 60 degrees, but too long.
Steg: Starting with "S" is good because then you get "circle, square, steg." Could be "streg," so you get the "tr" and "g" from "triangle."
Trueangle: OK, that's stupid.
Triddle: Goes nicely with "circle," and it looks like a real word, rather than something invented by a crank. This one has my vote, but I'm open to better suggestions.