Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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

\(\mathbf{2}\) \((020)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}\frac{2\sqrt{21}+2\sqrt{7}+\sqrt{3}+13}{18}\overrightarrow{AI},&\overrightarrow{BB^\prime}&={}-\frac{2\sqrt{21}+2\sqrt{7}-\sqrt{3}-13}{10}\overrightarrow{BI},&\overrightarrow{CC^\prime}&={}-\frac{2\sqrt{21}-2\sqrt{7}-\sqrt{3}-17}{2}\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{4\sqrt{21}+4\sqrt{7}+2\sqrt{3}-19}{45}&{}:{}&\frac{2\sqrt{21}+2\sqrt{7}+\sqrt{3}+13}{54}&{}:{}&\frac{7\left(2\sqrt{21}+2\sqrt{7}+\sqrt{3}+13\right)}{270}&,\\B^\prime&={}&-\frac{2\sqrt{21}+2\sqrt{7}-\sqrt{3}-13}{50}&{}:{}&\frac{2\sqrt{21}+2\sqrt{7}-\sqrt{3}+2}{15}&{}:{}&-\frac{7\left(2\sqrt{21}+2\sqrt{7}-\sqrt{3}-13\right)}{150}&,\\C^\prime&={}&-\frac{2\sqrt{21}-2\sqrt{7}-\sqrt{3}-17}{10}&{}:{}&-\frac{2\sqrt{21}-2\sqrt{7}-\sqrt{3}-17}{6}&{}:{}&\frac{8\sqrt{21}-8\sqrt{7}-4\sqrt{3}-53}{15}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}1.621594712201\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.027539679553\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}7.429201019893\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&-0.297275769760&{}:{}&0.540531570734&{}:{}&0.756744199027&,\\B^\prime&\approx{}&0.005507935911&{}:{}&0.981640213631&{}:{}&0.012851850458&,\\C^\prime&\approx{}&1.485840203979&{}:{}&2.476400339964&{}:{}&-2.962240543943&.\end{alignedat}\]
2 (020)

Hiroyasu Kamo