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]

\(\mathbf{4}\) \((200)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.987760142421&{}:{}&0.005099940658&{}:{}&0.007139916921&,\\B^\prime&{}\approx{}&0.583774096392&{}:{}&-0.945913654641&{}:{}&1.362139558248&,\\C^\prime&{}\approx{}&2.260569957535&{}:{}&3.767616595892&{}:{}&-5.028186553427&. \end{alignedat} \]
4 (200)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.015299821974\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}2.918870481961\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}11.302849787676\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&. \end{alignedat} \]
4 (200)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.958339177159&{}:{}&0.014798265296&{}:{}&0.026862557545&. \end{alignedat} \]
4 (200)

Hiroyasu Kamo