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]

\(\mathbf{5c}\) \((312)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&-21.186745878783&{}:{}&-55.466864696957&{}:{}&77.653610575740&,\\B^\prime&{}\approx{}&-0.504632472192&{}:{}&0.327156703744&{}:{}&1.177475768448&,\\C^\prime&{}\approx{}&-0.198129682738&{}:{}&-0.330216137896&{}:{}&1.528345820634&. \end{alignedat} \]
5c (312)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-11.093372939391\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-0.168210824064\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-0.066043227579\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&. \end{alignedat} \]
5c (312)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.566126464975&{}:{}&-0.576520425929&{}:{}&2.142646890904&. \end{alignedat} \]
5c (312)

Hiroyasu Kamo