A Generalized Malfatti Problem

Malfatti’s problem, first published in 1803, is commonly understood to ask fitting three circles into a given triangle such that they are tangent to each other, externally, and such that each circle is tangent to a pair of the triangles sides. There are many solutions based on geometric constructions, as well as generalizations in which the triangle sides are assumed to be circle arcs. A generalization that asks to fit six circles into the triangle, tangent to each other and to the triangle sides, has been considered a good example of a problem that requires sophisticated numerical iteration to solve by computer. We analyze this problem and show how to solve it quickly.

Ching-Shoei Chiang, Christoph M Hoffmann, and Paul Rosen. A Generalized Malfatti Problem. Computational Geometry: Theory and Applications, 2012.

@article{chiang2012generalized,
  title = {A Generalized Malfatti Problem},
  author = {Chiang, Ching-Shoei and Hoffmann, Christoph M and Rosen, Paul},
  journal = {Computational Geometry: Theory and Applications},
  volume = {45},
  pages = {425--435},
  year = {2012},
  keywords = {Malfatti's problem, circle packing, geometric constraint solving,
GPU programming},
  abstract = {Malfatti's problem, first published in 1803, is commonly understood to ask
    fitting three circles into a given triangle such that they are tangent to each other,
    externally, and such that each circle is tangent to a pair of the triangles sides. There
    are many solutions based on geometric constructions, as well as generalizations in which
    the triangle sides are assumed to be circle arcs. A generalization that asks to fit six
    circles into the triangle, tangent to each other and to the triangle sides, has been
    considered a good example of a problem that requires sophisticated numerical iteration
    to solve by computer. We analyze this problem and show how to solve it quickly.}
}