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**)

\(\mathbf{2a}\) \((031)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}\frac{4\sqrt{21}-5\sqrt{3}+9}{30}\overrightarrow{AI_A},&\overrightarrow{BB^\prime}&={}\frac{4\sqrt{21}+5\sqrt{3}-9}{2}\overrightarrow{BI_A},&\overrightarrow{CC^\prime}&={}\frac{8\sqrt{21}-5\sqrt{3}-27}{30}\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{8\sqrt{21}-10\sqrt{3}-27}{45}&{}:{}&\frac{4\sqrt{21}-5\sqrt{3}+9}{54}&{}:{}&\frac{7\left(4\sqrt{21}-5\sqrt{3}+9\right)}{270}&,\\B^\prime&={}&-\frac{4\sqrt{21}+5\sqrt{3}-9}{6}&{}:{}&-\frac{8\sqrt{21}+10\sqrt{3}-27}{9}&{}:{}&\frac{7\left(4\sqrt{21}+5\sqrt{3}-9\right)}{18}&,\\C^\prime&={}&-\frac{8\sqrt{21}-5\sqrt{3}-27}{90}&{}:{}&\frac{8\sqrt{21}-5\sqrt{3}-27}{54}&{}:{}&-\frac{8\sqrt{21}-5\sqrt{3}-162}{135}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.622334958066\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}8.995278408834\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}0.033345050727\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.170220055912&{}:{}&0.345741643370&{}:{}&0.484038300718&,\\B^\prime&\approx{}&-2.998426136278&{}:{}&-2.997901515037&{}:{}&6.996327651315&,\\C^\prime&\approx{}&-0.011115016909&{}:{}&0.018525028182&{}:{}&0.992589988727&.\end{alignedat}\]
2a (031)

Hiroyasu Kamo