Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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\(\mathbf{4c}\) \((310)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}6.629510156769\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}0.075420353567\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}0.258469396840\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&4.314755078384&{}:{}&13.259020313537&{}:{}&-16.573775391922&,\\B^\prime&\approx{}&0.113130530351&{}:{}&1.075420353567&{}:{}&-0.188550883918&,\\C^\prime&\approx{}&0.387704095259&{}:{}&0.516938793679&{}:{}&0.095357111061&.\end{alignedat}\]
4c (310)

Hiroyasu Kamo