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]

\(\mathbf{5}\) \((202)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.963942436427&{}:{}&0.015023984822&{}:{}&0.021033578751&,\\B^\prime&{}\approx{}&0.594492064089&{}:{}&-0.981640213631&{}:{}&1.387148149542&,\\C^\prime&{}\approx{}&1.514159796021&{}:{}&2.523599660036&{}:{}&-3.037759456057&. \end{alignedat} \]
5 (202)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.045071954466\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}2.972460320447\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}7.570798980107\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&. \end{alignedat} \]
5 (202)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.909477399214&{}:{}&0.014043762537&{}:{}&0.076478838249&. \end{alignedat} \]
5 (202)

Hiroyasu Kamo