Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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[Lob & Richmond]
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\(\mathbf{1a}\) \((013)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.659742872299\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}0.130030109045\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}2.597262605646\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.010385691552&{}:{}&0.439828581532&{}:{}&0.549785726915&,\\B^\prime&\approx{}&-0.065015054523&{}:{}&0.956656630318&{}:{}&0.108358424204&,\\C^\prime&\approx{}&-1.298631302823&{}:{}&1.731508403764&{}:{}&0.567122899059&.\end{alignedat}\]
1a (013)

Hiroyasu Kamo