Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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\(\mathbf{2c}\) \((130)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.113130530351\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}4.419673437846\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}0.322410832200\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&1.056565265175&{}:{}&0.226261060701&{}:{}&-0.282826325877&,\\B^\prime&\approx{}&6.629510156769&{}:{}&5.419673437846&{}:{}&-11.049183594614&,\\C^\prime&\approx{}&0.483616248300&{}:{}&0.644821664400&{}:{}&-0.128437912700&.\end{alignedat}\]
2c (130)

Hiroyasu Kamo