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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\(\mathbf{4b}\) \((301)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}7.556561139894\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}0.385653408796\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}0.050547597535\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&-0.889140284973&{}:{}&-7.556561139894&{}:{}&9.445701424867&,\\B^\prime&\approx{}&0.289240056597&{}:{}&0.228693182409&{}:{}&0.482066760995&,\\C^\prime&\approx{}&0.037910698151&{}:{}&-0.050547597535&{}:{}&1.012636899384&.\end{alignedat}\]
4b (301)

Hiroyasu Kamo