Start with a blue triangle, and form the green triangle whose vertices bisect each circular arc connecting blue vertices. Similarly, make the red triangle from the green, and the orange triangle from the red. Prove the triangle becomes equilateral in the limit.
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Name the angles in the triangle a, b and c. Then the first iteration is stepping to (b+c)/2, (c+a)/2 and (a+b)/2. But b+c = 180-a. So iteration is really just x -> 90 -(x/2). Or equivalently 60-x -> (60-x)/2. So every angle converges to 60.