Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.751999353383\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.664894188740\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}0.507933962352\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.436000484963&{}:{}&0.250666451128&{}:{}&0.313333063909&,\\B^\prime&\approx{}&0.166223547185&{}:{}&0.556737207507&{}:{}&0.277039245308&,\\C^\prime&\approx{}&0.126983490588&{}:{}&0.169311320784&{}:{}&0.703705188628&.\end{alignedat}\]
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Hiroyasu Kamo