I tried linking the points of an equilateral triangle with arc (using OnShape, which is the program I know for playing with geometry...), and it *seems* like I can get three circular arcs happening that aren't equal. Are they secretly overlapping a tiny tiny bit if they aren't all mutually tangent? Or something? (Took a screenshot but can't link it in the comments...)

OK, after a few false starts I think I understand this. For any two points there are infinitely many circles that pass through them. So we can TOTALLY get three circular arcs that aren't equal through three points. But I think with the further restriction that those arcs are part of mutually tangent circles, I THINK that the arcs about an equilateral triangle have to be equal. I think...?

Ahhhh, yup. I tried with full circles this time and the problem becomes clear! If the circles aren't mutually tangent then there's overlap happening somewhere.

I tried linking the points of an equilateral triangle with arc (using OnShape, which is the program I know for playing with geometry...), and it *seems* like I can get three circular arcs happening that aren't equal. Are they secretly overlapping a tiny tiny bit if they aren't all mutually tangent? Or something? (Took a screenshot but can't link it in the comments...)

OK, after a few false starts I think I understand this. For any two points there are infinitely many circles that pass through them. So we can TOTALLY get three circular arcs that aren't equal through three points. But I think with the further restriction that those arcs are part of mutually tangent circles, I THINK that the arcs about an equilateral triangle have to be equal. I think...?

Ahhhh, yup. I tried with full circles this time and the problem becomes clear! If the circles aren't mutually tangent then there's overlap happening somewhere.