Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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\(\mathbf{4a}\) \((211)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}-\frac{2\sqrt{10}+\sqrt{5}+2\sqrt{2}-1}{12}\overrightarrow{AI_A},&\overrightarrow{BB^\prime}&={}-\frac{2\sqrt{10}+\sqrt{5}-2\sqrt{2}+1}{2}\overrightarrow{BI_A},&\overrightarrow{CC^\prime}&={}-\frac{2\sqrt{10}-\sqrt{5}+2\sqrt{2}+1}{4}\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&\frac{2\sqrt{10}+\sqrt{5}+2\sqrt{2}+7}{8}&{}:{}&-\frac{2\sqrt{10}+\sqrt{5}+2\sqrt{2}-1}{18}&{}:{}&-\frac{5\left(2\sqrt{10}+\sqrt{5}+2\sqrt{2}-1\right)}{72}&,\\B^\prime&={}&\frac{2\sqrt{10}+\sqrt{5}-2\sqrt{2}+1}{4}&{}:{}&\frac{2\sqrt{10}+\sqrt{5}-2\sqrt{2}+7}{6}&{}:{}&-\frac{5\left(2\sqrt{10}+\sqrt{5}-2\sqrt{2}+1\right)}{12}&,\\C^\prime&={}&\frac{2\sqrt{10}-\sqrt{5}+2\sqrt{2}+1}{8}&{}:{}&-\frac{2\sqrt{10}-\sqrt{5}+2\sqrt{2}+1}{6}&{}:{}&\frac{2\sqrt{10}-\sqrt{5}+2\sqrt{2}+25}{24}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-0.865754201882\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-3.366098086545\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-1.979228616896\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&2.298631302823&{}:{}&-0.577169467921&{}:{}&-0.721461834902&,\\B^\prime&\approx{}&1.683049043273&{}:{}&2.122032695515&{}:{}&-2.805081738788&,\\C^\prime&\approx{}&0.989614308448&{}:{}&-1.319485744597&{}:{}&1.329871436149&.\end{alignedat}\]
4a (211)

Hiroyasu Kamo