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]

\(\mathbf{0b}\) \((101)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.884489355921&{}:{}&-0.288776610198&{}:{}&0.404287254278&,\\B^\prime&{}\approx{}&0.835089355133&{}:{}&-1.783631183776&{}:{}&1.948541828643&,\\C^\prime&{}\approx{}&0.359305255565&{}:{}&-0.598842092609&{}:{}&1.239536837044&. \end{alignedat} \]
0b (101)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.288776610198\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}1.391815591888\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}0.598842092609\overrightarrow{CI_B}. \end{aligned} \] \[ \begin{alignedat}{4} I_B&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&. \end{alignedat} \]
0b (101)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.681265979294&{}:{}&-0.694130366975&{}:{}&1.012864387681&. \end{alignedat} \]
0b (101)

Hiroyasu Kamo