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]

\(\mathbf{1}\) \((002)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&-0.004697758905&{}:{}&0.418624066210&{}:{}&0.586073692694&,\\B^\prime&{}\approx{}&0.297168040796&{}:{}&0.009439864014&{}:{}&0.693392095190&,\\C^\prime&{}\approx{}&0.027539679553&{}:{}&0.045899465921&{}:{}&0.926560854526&. \end{alignedat} \]
1 (002)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}1.255872198631\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}1.485840203979\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}0.137698397764\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&. \end{alignedat} \]
1 (002)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.083698291999&{}:{}&0.142155920044&{}:{}&0.774145787956&. \end{alignedat} \]
1 (002)

Hiroyasu Kamo