Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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\(\mathbf{2}\) \((020)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}\frac{\sqrt{10}+\sqrt{5}-\sqrt{2}+7}{6}\overrightarrow{AI},&\overrightarrow{BB^\prime}&={}-\frac{\sqrt{10}+\sqrt{5}+\sqrt{2}-7}{4}\overrightarrow{BI},&\overrightarrow{CC^\prime}&={}-\frac{\sqrt{10}-\sqrt{5}-\sqrt{2}-7}{2}\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{\sqrt{10}+\sqrt{5}-\sqrt{2}-1}{8}&{}:{}&\frac{\sqrt{10}+\sqrt{5}-\sqrt{2}+7}{18}&{}:{}&\frac{5\left(\sqrt{10}+\sqrt{5}-\sqrt{2}+7\right)}{72}&,\\B^\prime&={}&-\frac{\sqrt{10}+\sqrt{5}+\sqrt{2}-7}{16}&{}:{}&\frac{\sqrt{10}+\sqrt{5}+\sqrt{2}-1}{6}&{}:{}&-\frac{5\left(\sqrt{10}+\sqrt{5}+\sqrt{2}-7\right)}{48}&,\\C^\prime&={}&-\frac{\sqrt{10}-\sqrt{5}-\sqrt{2}-7}{8}&{}:{}&-\frac{\sqrt{10}-\sqrt{5}-\sqrt{2}-7}{6}&{}:{}&\frac{7\sqrt{10}-7\sqrt{5}-7\sqrt{2}-25}{24}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}1.830688679216\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.046860199990\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}3.744001939852\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&-0.373016509412&{}:{}&0.610229559739&{}:{}&0.762786949673&,\\B^\prime&\approx{}&0.011715049997&{}:{}&0.968759866674&{}:{}&0.019525083329&,\\C^\prime&\approx{}&0.936000484963&{}:{}&1.248000646617&{}:{}&-1.184001131580&.\end{alignedat}\]
2 (020)

Hiroyasu Kamo