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]

\(\mathbf{4a}\) \((211)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.799580303007&{}:{}&-0.333158459586&{}:{}&-0.466421843421&,\\B^\prime&{}\approx{}&3.111674790330&{}:{}&5.148899720440&{}:{}&-7.260574510770&,\\C^\prime&{}\approx{}&0.803565106639&{}:{}&-1.339275177731&{}:{}&1.535710071093&. \end{alignedat} \]
4a (211)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-0.599685227256\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-9.335024370989\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-2.410695319916\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&. \end{alignedat} \]
4a (211)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&1.558793627719&{}:{}&-0.069594650621&{}:{}&-0.489198977098&. \end{alignedat} \]
4a (211)

Hiroyasu Kamo