Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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[Lob & Richmond]
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\(\mathbf{6a}\) \((231)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}-\frac{2\sqrt{10}+\sqrt{5}-2\sqrt{2}-5}{12}\overrightarrow{AI_A},&\overrightarrow{BB^\prime}&={}-\frac{2\sqrt{10}+\sqrt{5}+2\sqrt{2}+5}{2}\overrightarrow{BI_A},&\overrightarrow{CC^\prime}&={}-\frac{2\sqrt{10}-\sqrt{5}-2\sqrt{2}+5}{4}\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&\frac{2\sqrt{10}+\sqrt{5}-2\sqrt{2}+3}{8}&{}:{}&-\frac{2\sqrt{10}+\sqrt{5}-2\sqrt{2}-5}{18}&{}:{}&-\frac{5\left(2\sqrt{10}+\sqrt{5}-2\sqrt{2}-5\right)}{72}&,\\B^\prime&={}&\frac{2\sqrt{10}+\sqrt{5}+2\sqrt{2}+5}{4}&{}:{}&\frac{2\sqrt{10}+\sqrt{5}+2\sqrt{2}+11}{6}&{}:{}&-\frac{5\left(2\sqrt{10}+\sqrt{5}+2\sqrt{2}+5\right)}{12}&,\\C^\prime&={}&\frac{2\sqrt{10}-\sqrt{5}-2\sqrt{2}+5}{8}&{}:{}&-\frac{2\sqrt{10}-\sqrt{5}-2\sqrt{2}+5}{6}&{}:{}&\frac{2\sqrt{10}-\sqrt{5}-2\sqrt{2}+29}{24}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-0.061016347758\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-8.194525211291\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-1.565015054523\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&1.091524521636&{}:{}&-0.040677565172&{}:{}&-0.050846956465&,\\B^\prime&\approx{}&4.097262605646&{}:{}&3.731508403764&{}:{}&-6.828771009409&,\\C^\prime&\approx{}&0.782507527261&{}:{}&-1.043343369682&{}:{}&1.260835842420&.\end{alignedat}\]
6a (231)

Hiroyasu Kamo