Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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\(\mathbf{3}\) \((022)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}1.968759866674\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.253966981176\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}1.329788377479\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&-0.476569900005&{}:{}&0.656253288891&{}:{}&0.820316611114&,\\B^\prime&\approx{}&0.063491745294&{}:{}&0.830688679216&{}:{}&0.105819575490&,\\C^\prime&\approx{}&0.332447094370&{}:{}&0.443262792493&{}:{}&0.224290113137&.\end{alignedat}\]
3 (022)

Hiroyasu Kamo