Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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\(\mathbf{5a}\) \((213)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}-\frac{2\sqrt{10}-\sqrt{5}+2\sqrt{2}-5}{12}\overrightarrow{AI_A},&\overrightarrow{BB^\prime}&={}-\frac{2\sqrt{10}-\sqrt{5}-2\sqrt{2}+5}{2}\overrightarrow{BI_A},&\overrightarrow{CC^\prime}&={}-\frac{2\sqrt{10}+\sqrt{5}+2\sqrt{2}+5}{4}\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&\frac{2\sqrt{10}-\sqrt{5}+2\sqrt{2}+3}{8}&{}:{}&-\frac{2\sqrt{10}-\sqrt{5}+2\sqrt{2}-5}{18}&{}:{}&-\frac{5\left(2\sqrt{10}-\sqrt{5}+2\sqrt{2}-5\right)}{72}&,\\B^\prime&={}&\frac{2\sqrt{10}-\sqrt{5}-2\sqrt{2}+5}{4}&{}:{}&\frac{2\sqrt{10}-\sqrt{5}-2\sqrt{2}+11}{6}&{}:{}&-\frac{5\left(2\sqrt{10}-\sqrt{5}-2\sqrt{2}+5\right)}{12}&,\\C^\prime&={}&\frac{2\sqrt{10}+\sqrt{5}+2\sqrt{2}+5}{8}&{}:{}&-\frac{2\sqrt{10}+\sqrt{5}+2\sqrt{2}+5}{6}&{}:{}&\frac{2\sqrt{10}+\sqrt{5}+2\sqrt{2}+29}{24}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-0.159742872299\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-3.130030109045\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-4.097262605646\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&1.239614308448&{}:{}&-0.106495248199&{}:{}&-0.133119060249&,\\B^\prime&\approx{}&1.565015054523&{}:{}&2.043343369682&{}:{}&-2.608358424204&,\\C^\prime&\approx{}&2.048631302823&{}:{}&-2.731508403764&{}:{}&1.682877100941&.\end{alignedat}\]
5a (213)

Hiroyasu Kamo