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]

\(\mathbf{3b}\) \((123)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&3.077370564310&{}:{}&5.193426410776&{}:{}&-7.270796975086&,\\B^\prime&{}\approx{}&-0.215583153339&{}:{}&1.718610511131&{}:{}&-0.503027357792&,\\C^\prime&{}\approx{}&-1.391815591888&{}:{}&2.319692653146&{}:{}&0.072122938741&. \end{alignedat} \]
3b (123)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-5.193426410776\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}-0.359305255565\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}-2.319692653146\overrightarrow{CI_B}. \end{aligned} \] \[ \begin{alignedat}{4} I_B&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&. \end{alignedat} \]
3b (123)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.070106174355&{}:{}&2.521836547934&{}:{}&-1.451730373579&. \end{alignedat} \]
3b (123)

Hiroyasu Kamo