Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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\(\mathbf{6}\) \((220)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}-\frac{\sqrt{10}-\sqrt{5}+\sqrt{2}-5}{6}\overrightarrow{AI},&\overrightarrow{BB^\prime}&={}\frac{\sqrt{10}-\sqrt{5}-\sqrt{2}+5}{4}\overrightarrow{BI},&\overrightarrow{CC^\prime}&={}\frac{\sqrt{10}+\sqrt{5}+\sqrt{2}+5}{2}\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&\frac{\sqrt{10}-\sqrt{5}+\sqrt{2}+3}{8}&{}:{}&-\frac{\sqrt{10}-\sqrt{5}+\sqrt{2}-5}{18}&{}:{}&-\frac{5\left(\sqrt{10}-\sqrt{5}+\sqrt{2}-5\right)}{72}&,\\B^\prime&={}&\frac{\sqrt{10}-\sqrt{5}-\sqrt{2}+5}{16}&{}:{}&-\frac{\sqrt{10}-\sqrt{5}-\sqrt{2}-1}{6}&{}:{}&\frac{5\left(\sqrt{10}-\sqrt{5}-\sqrt{2}+5\right)}{48}&,\\C^\prime&={}&\frac{\sqrt{10}+\sqrt{5}+\sqrt{2}+5}{8}&{}:{}&\frac{\sqrt{10}+\sqrt{5}+\sqrt{2}+5}{6}&{}:{}&-\frac{7\sqrt{10}+7\sqrt{5}+7\sqrt{2}+11}{24}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.443262792493\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}1.127999030074\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}5.906279600021\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.667552905630&{}:{}&0.147754264164&{}:{}&0.184692830205&,\\B^\prime&\approx{}&0.281999757518&{}:{}&0.248000646617&{}:{}&0.469999595864&,\\C^\prime&\approx{}&1.476569900005&{}:{}&1.968759866674&{}:{}&-2.445329766679&.\end{alignedat}\]
6 (220)

Hiroyasu Kamo