Repeatedly bisecting the largest angle of any nondegenerate triangle makes the pieces shrink to zero, keeps the smallest angle bounded away from zero, and generates infinitely many similarity types unless the start is an isosceles right triangle.
Stynes, On faster convergence of the bisection method for all triangles
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The largest angle bisection procedure
Repeatedly bisecting the largest angle of any nondegenerate triangle makes the pieces shrink to zero, keeps the smallest angle bounded away from zero, and generates infinitely many similarity types unless the start is an isosceles right triangle.