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]

\(\mathbf{3a}\) \((033)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.185179977455&{}:{}&0.339508342727&{}:{}&0.475311679818&,\\B^\prime&{}\approx{}&-1.017825533194&{}:{}&-0.357100710925&{}:{}&2.374926244119&,\\C^\prime&{}\approx{}&-0.022334958066&{}:{}&0.037224930110&{}:{}&0.985110027956&. \end{alignedat} \]
3a (033)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.611115016909\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}3.053476599582\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}0.067004874198\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&. \end{alignedat} \]
3a (033)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.026021777990&{}:{}&0.127785328297&{}:{}&0.898236449692&. \end{alignedat} \]
3a (033)

Hiroyasu Kamo