Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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\(\mathbf{3b}\) \((123)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}\frac{\sqrt{10}-3\sqrt{5}+3\sqrt{2}-5}{2}\overrightarrow{AI_B},&\overrightarrow{BB^\prime}&={}\frac{\sqrt{10}-3\sqrt{5}-3\sqrt{2}+5}{12}\overrightarrow{BI_B},&\overrightarrow{CC^\prime}&={}-\frac{\sqrt{10}+3\sqrt{5}+3\sqrt{2}+5}{6}\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{\sqrt{10}-3\sqrt{5}+3\sqrt{2}-13}{8}&{}:{}&-\frac{\sqrt{10}-3\sqrt{5}+3\sqrt{2}-5}{2}&{}:{}&\frac{5\left(\sqrt{10}-3\sqrt{5}+3\sqrt{2}-5\right)}{8}&,\\B^\prime&={}&\frac{\sqrt{10}-3\sqrt{5}-3\sqrt{2}+5}{16}&{}:{}&-\frac{\sqrt{10}-3\sqrt{5}-3\sqrt{2}-1}{6}&{}:{}&\frac{5\left(\sqrt{10}-3\sqrt{5}-3\sqrt{2}+5\right)}{48}&,\\C^\prime&={}&-\frac{\sqrt{10}+3\sqrt{5}+3\sqrt{2}+5}{8}&{}:{}&\frac{\sqrt{10}+3\sqrt{5}+3\sqrt{2}+5}{6}&{}:{}&-\frac{\sqrt{10}+3\sqrt{5}+3\sqrt{2}-19}{24}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-2.151642792606\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}-0.232380579954\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}-3.185520379965\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&1.537910698151&{}:{}&2.151642792606&{}:{}&-2.689553490757&,\\B^\prime&\approx{}&-0.174285434966&{}:{}&1.464761159908&{}:{}&-0.290475724943&,\\C^\prime&\approx{}&-2.389140284973&{}:{}&3.185520379965&{}:{}&0.203619905009&.\end{alignedat}\]
3b (123)

Hiroyasu Kamo