Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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\(\mathbf{5c}\) \((312)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}-\frac{2\sqrt{10}+3\sqrt{5}+6\sqrt{2}+7}{4}\overrightarrow{AI_C},&\overrightarrow{BB^\prime}&={}\frac{2\sqrt{10}+3\sqrt{5}-6\sqrt{2}-7}{6}\overrightarrow{BI_C},&\overrightarrow{CC^\prime}&={}-\frac{2\sqrt{10}-3\sqrt{5}+6\sqrt{2}-7}{12}\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{2\sqrt{10}+3\sqrt{5}+6\sqrt{2}-1}{8}&{}:{}&-\frac{2\sqrt{10}+3\sqrt{5}+6\sqrt{2}+7}{2}&{}:{}&\frac{5\left(2\sqrt{10}+3\sqrt{5}+6\sqrt{2}+7\right)}{8}&,\\B^\prime&={}&\frac{2\sqrt{10}+3\sqrt{5}-6\sqrt{2}-7}{4}&{}:{}&\frac{2\sqrt{10}+3\sqrt{5}-6\sqrt{2}-1}{6}&{}:{}&-\frac{5\left(2\sqrt{10}+3\sqrt{5}-6\sqrt{2}-7\right)}{12}&,\\C^\prime&={}&-\frac{2\sqrt{10}-3\sqrt{5}+6\sqrt{2}-7}{8}&{}:{}&-\frac{2\sqrt{10}-3\sqrt{5}+6\sqrt{2}-7}{6}&{}:{}&\frac{14\sqrt{10}-21\sqrt{5}+42\sqrt{2}-25}{24}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-7.129510156769\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-0.408753686900\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-0.091802730173\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&-2.564755078384&{}:{}&-14.259020313537&{}:{}&17.823775391922&,\\B^\prime&\approx{}&-0.613130530351&{}:{}&0.591246313100&{}:{}&1.021884217251&,\\C^\prime&\approx{}&-0.137704095259&{}:{}&-0.183605460346&{}:{}&1.321309555605&.\end{alignedat}\]
5c (312)

Hiroyasu Kamo