Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


[Other solutions]
[Guy]
[Lob & Richmond]
(0**)
(1**)
(2**)
(3**)

\(\mathbf{7a}\) \((233)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}-\frac{2\sqrt{10}-\sqrt{5}-2\sqrt{2}-1}{12}\overrightarrow{AI_A},&\overrightarrow{BB^\prime}&={}-\frac{2\sqrt{10}-\sqrt{5}+2\sqrt{2}+1}{2}\overrightarrow{BI_A},&\overrightarrow{CC^\prime}&={}-\frac{2\sqrt{10}+\sqrt{5}-2\sqrt{2}+1}{4}\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&\frac{2\sqrt{10}-\sqrt{5}-2\sqrt{2}+7}{8}&{}:{}&-\frac{2\sqrt{10}-\sqrt{5}-2\sqrt{2}-1}{18}&{}:{}&-\frac{5\left(2\sqrt{10}-\sqrt{5}-2\sqrt{2}-1\right)}{72}&,\\B^\prime&={}&\frac{2\sqrt{10}-\sqrt{5}+2\sqrt{2}+1}{4}&{}:{}&\frac{2\sqrt{10}-\sqrt{5}+2\sqrt{2}+7}{6}&{}:{}&-\frac{5\left(2\sqrt{10}-\sqrt{5}+2\sqrt{2}+1\right)}{12}&,\\C^\prime&={}&\frac{2\sqrt{10}+\sqrt{5}-2\sqrt{2}+1}{8}&{}:{}&-\frac{2\sqrt{10}+\sqrt{5}-2\sqrt{2}+1}{6}&{}:{}&\frac{2\sqrt{10}+\sqrt{5}-2\sqrt{2}+25}{24}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-0.021671684841\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-3.958457233792\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-1.683049043273\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&1.032507527261&{}:{}&-0.014447789894&{}:{}&-0.018059737367&,\\B^\prime&\approx{}&1.979228616896&{}:{}&2.319485744597&{}:{}&-3.298714361493&,\\C^\prime&\approx{}&0.841524521636&{}:{}&-1.122032695515&{}:{}&1.280508173879&.\end{alignedat}\]
7a (233)

Hiroyasu Kamo