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]

\(\mathbf{0c}\) \((110)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.396259365476&{}:{}&0.990648413689&{}:{}&-1.386907779164&,\\B^\prime&{}\approx{}&0.676196490926&{}:{}&1.901595321235&{}:{}&-1.577791812161&,\\C^\prime&{}\approx{}&11.093372939391&{}:{}&18.488954898986&{}:{}&-28.582327838377&. \end{alignedat} \]
0c (110)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.198129682738\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}0.225398830309\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}3.697790979797\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&. \end{alignedat} \]
0c (110)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&1.554181635732&{}:{}&2.320320697670&{}:{}&-2.874502333401&. \end{alignedat} \]
0c (110)

Hiroyasu Kamo