Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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\(\mathbf{7}\) \((222)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}1.248000646617\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}2.335105811260\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}5.492066037648\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.063999515037&{}:{}&0.416000215539&{}:{}&0.520000269424&,\\B^\prime&\approx{}&0.583776452815&{}:{}&-0.556737207507&{}:{}&0.972960754692&,\\C^\prime&\approx{}&1.373016509412&{}:{}&1.830688679216&{}:{}&-2.203705188628&.\end{alignedat}\]
7 (222)

Hiroyasu Kamo