Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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\(\mathbf{7b}\) \((323)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-3.394283479725\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}-0.025273798768\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}-0.771306817591\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&1.848570869931&{}:{}&3.394283479725&{}:{}&-4.242854349656&,\\B^\prime&\approx{}&-0.018955349076&{}:{}&1.050547597535&{}:{}&-0.031592248460&,\\C^\prime&\approx{}&-0.578480113194&{}:{}&0.771306817591&{}:{}&0.807173295602&.\end{alignedat}\]
7b (323)

Hiroyasu Kamo