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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\(\mathbf{1b}\) \((103)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}-\frac{\sqrt{10}-3\sqrt{5}+3\sqrt{2}-1}{2}\overrightarrow{AI_B},&\overrightarrow{BB^\prime}&={}-\frac{\sqrt{10}-3\sqrt{5}-3\sqrt{2}+1}{12}\overrightarrow{BI_B},&\overrightarrow{CC^\prime}&={}\frac{\sqrt{10}+3\sqrt{5}+3\sqrt{2}+1}{6}\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&\frac{\sqrt{10}-3\sqrt{5}+3\sqrt{2}+7}{8}&{}:{}&\frac{\sqrt{10}-3\sqrt{5}+3\sqrt{2}-1}{2}&{}:{}&-\frac{5\left(\sqrt{10}-3\sqrt{5}+3\sqrt{2}-1\right)}{8}&,\\B^\prime&={}&-\frac{\sqrt{10}-3\sqrt{5}-3\sqrt{2}+1}{16}&{}:{}&\frac{\sqrt{10}-3\sqrt{5}-3\sqrt{2}+7}{6}&{}:{}&-\frac{5\left(\sqrt{10}-3\sqrt{5}-3\sqrt{2}+1\right)}{48}&,\\C^\prime&={}&\frac{\sqrt{10}+3\sqrt{5}+3\sqrt{2}+1}{8}&{}:{}&-\frac{\sqrt{10}+3\sqrt{5}+3\sqrt{2}+1}{6}&{}:{}&\frac{\sqrt{10}+3\sqrt{5}+3\sqrt{2}+25}{24}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.151642792606\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}0.565713913288\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}2.518853713298\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.962089301849&{}:{}&-0.151642792606&{}:{}&0.189553490757&,\\B^\prime&\approx{}&0.424285434966&{}:{}&-0.131427826575&{}:{}&0.707142391609&,\\C^\prime&\approx{}&1.889140284973&{}:{}&-2.518853713298&{}:{}&1.629713428324&.\end{alignedat}\]
1b (103)

Hiroyasu Kamo