Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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\(\mathbf{2b}\) \((121)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}-\frac{\sqrt{10}+3\sqrt{5}-3\sqrt{2}-1}{2}\overrightarrow{AI_B},&\overrightarrow{BB^\prime}&={}-\frac{\sqrt{10}+3\sqrt{5}+3\sqrt{2}+1}{12}\overrightarrow{BI_B},&\overrightarrow{CC^\prime}&={}\frac{\sqrt{10}-3\sqrt{5}-3\sqrt{2}+1}{6}\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&\frac{\sqrt{10}+3\sqrt{5}-3\sqrt{2}+7}{8}&{}:{}&\frac{\sqrt{10}+3\sqrt{5}-3\sqrt{2}-1}{2}&{}:{}&-\frac{5\left(\sqrt{10}+3\sqrt{5}-3\sqrt{2}-1\right)}{8}&,\\B^\prime&={}&-\frac{\sqrt{10}+3\sqrt{5}+3\sqrt{2}+1}{16}&{}:{}&\frac{\sqrt{10}+3\sqrt{5}+3\sqrt{2}+7}{6}&{}:{}&-\frac{5\left(\sqrt{10}+3\sqrt{5}+3\sqrt{2}+1\right)}{48}&,\\C^\prime&={}&\frac{\sqrt{10}-3\sqrt{5}-3\sqrt{2}+1}{8}&{}:{}&-\frac{\sqrt{10}-3\sqrt{5}-3\sqrt{2}+1}{6}&{}:{}&\frac{\sqrt{10}-3\sqrt{5}-3\sqrt{2}+25}{24}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-2.313920452774\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}-1.259426856649\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}-1.131427826575\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&1.578480113194&{}:{}&2.313920452774&{}:{}&-2.892400565968&,\\B^\prime&\approx{}&-0.944570142487&{}:{}&3.518853713298&{}:{}&-1.574283570811&,\\C^\prime&\approx{}&-0.848570869931&{}:{}&1.131427826575&{}:{}&0.717143043356&.\end{alignedat}\]
2b (121)

Hiroyasu Kamo