Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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[Guy]
[Lob & Richmond]
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\(\mathbf{3a}\) \((033)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}\frac{2\sqrt{10}-\sqrt{5}-2\sqrt{2}+5}{12}\overrightarrow{AI_A},&\overrightarrow{BB^\prime}&={}\frac{2\sqrt{10}-\sqrt{5}+2\sqrt{2}-5}{2}\overrightarrow{BI_A},&\overrightarrow{CC^\prime}&={}\frac{2\sqrt{10}+\sqrt{5}-2\sqrt{2}-5}{4}\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{2\sqrt{10}-\sqrt{5}-2\sqrt{2}-3}{8}&{}:{}&\frac{2\sqrt{10}-\sqrt{5}-2\sqrt{2}+5}{18}&{}:{}&\frac{5\left(2\sqrt{10}-\sqrt{5}-2\sqrt{2}+5\right)}{72}&,\\B^\prime&={}&-\frac{2\sqrt{10}-\sqrt{5}+2\sqrt{2}-5}{4}&{}:{}&-\frac{2\sqrt{10}-\sqrt{5}+2\sqrt{2}-11}{6}&{}:{}&\frac{5\left(2\sqrt{10}-\sqrt{5}+2\sqrt{2}-5\right)}{12}&,\\C^\prime&={}&-\frac{2\sqrt{10}+\sqrt{5}-2\sqrt{2}-5}{8}&{}:{}&\frac{2\sqrt{10}+\sqrt{5}-2\sqrt{2}-5}{6}&{}:{}&-\frac{2\sqrt{10}+\sqrt{5}-2\sqrt{2}-29}{24}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.521671684841\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}0.958457233792\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}0.183049043273\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.217492472739&{}:{}&0.347781123227&{}:{}&0.434726404034&,\\B^\prime&\approx{}&-0.479228616896&{}:{}&0.680514255403&{}:{}&0.798714361493&,\\C^\prime&\approx{}&-0.091524521636&{}:{}&0.122032695515&{}:{}&0.969491826121&.\end{alignedat}\]
3a (033)

Hiroyasu Kamo