This fractal idea did not originate with me, but I wrote some racket code to do the midpoint calculation using barycentric coordinates. This fractal draws a circle at the midpoint of a triangle, then subdivides the triangle and repeats:

Here is the same fractal to four iterations:

To get the midpoints, I could simple pass in the coordinates of the last triangle ABC, and then use “0.5” barycentric coordinates:

[P1 (barycentric->complex 0.5 0.5 0 A B C)]
[P2 (barycentric->complex 0.5 0.0 0.5 A B C)]
[P3 (barycentric->complex 0 0.5 0.5 A B C)]