Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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\(\mathbf{1c}\) \((112)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}-\frac{2\sqrt{10}-3\sqrt{5}+6\sqrt{2}-5}{4}\overrightarrow{AI_C},&\overrightarrow{BB^\prime}&={}\frac{2\sqrt{10}-3\sqrt{5}-6\sqrt{2}+5}{6}\overrightarrow{BI_C},&\overrightarrow{CC^\prime}&={}-\frac{2\sqrt{10}+3\sqrt{5}+6\sqrt{2}+5}{12}\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{2\sqrt{10}-3\sqrt{5}+6\sqrt{2}-13}{8}&{}:{}&-\frac{2\sqrt{10}-3\sqrt{5}+6\sqrt{2}-5}{2}&{}:{}&\frac{5\left(2\sqrt{10}-3\sqrt{5}+6\sqrt{2}-5\right)}{8}&,\\B^\prime&={}&\frac{2\sqrt{10}-3\sqrt{5}-6\sqrt{2}+5}{4}&{}:{}&\frac{2\sqrt{10}-3\sqrt{5}-6\sqrt{2}+11}{6}&{}:{}&-\frac{5\left(2\sqrt{10}-3\sqrt{5}-6\sqrt{2}+5\right)}{12}&,\\C^\prime&={}&-\frac{2\sqrt{10}+3\sqrt{5}+6\sqrt{2}+5}{8}&{}:{}&-\frac{2\sqrt{10}+3\sqrt{5}+6\sqrt{2}+5}{6}&{}:{}&\frac{14\sqrt{10}+21\sqrt{5}+42\sqrt{2}+59}{24}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-0.775408190519\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-0.644821664400\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-2.209836718923\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.612295904741&{}:{}&-1.550816381038&{}:{}&1.938520476297&,\\B^\prime&\approx{}&-0.967232496600&{}:{}&0.355178335600&{}:{}&1.612054161000&,\\C^\prime&\approx{}&-3.314755078384&{}:{}&-4.419673437846&{}:{}&8.734428516230&.\end{alignedat}\]
1c (112)

Hiroyasu Kamo