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]

\(\mathbf{2c}\) \((130)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.205558966630&{}:{}&0.513897416576&{}:{}&-0.719456383207&,\\B^\prime&{}\approx{}&18.155621565652&{}:{}&25.207495420870&{}:{}&-42.363116986522&,\\C^\prime&{}\approx{}&0.605717894556&{}:{}&1.009529824259&{}:{}&-0.615247718815&. \end{alignedat} \]
2c (130)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.102779483315\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}6.051873855217\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}0.201905964852\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&. \end{alignedat} \]
2c (130)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&1.321393924846&{}:{}&1.345654435015&{}:{}&-1.667048359862&. \end{alignedat} \]
2c (130)

Hiroyasu Kamo