Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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\(\mathbf{7c}\) \((332)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-0.967232496600\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-0.516938793679\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-0.037710176784\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.516383751700&{}:{}&-1.934464993201&{}:{}&2.418081241501&,\\B^\prime&\approx{}&-0.775408190519&{}:{}&0.483061206321&{}:{}&1.292346984198&,\\C^\prime&\approx{}&-0.056565265175&{}:{}&-0.075420353567&{}:{}&1.131985618742&.\end{alignedat}\]
7c (332)

Hiroyasu Kamo