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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\(\mathbf{0c}\) \((110)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.275408190519\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}0.311488331067\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}2.376503385590\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&1.137704095259&{}:{}&0.550816381038&{}:{}&-0.688520476297&,\\B^\prime&\approx{}&0.467232496600&{}:{}&1.311488331067&{}:{}&-0.778720827667&,\\C^\prime&\approx{}&3.564755078384&{}:{}&4.753006771179&{}:{}&-7.317761849563&.\end{alignedat}\]
0c (110)

Hiroyasu Kamo