Derousseau's Generalization of the Malfatti circles

Equilateral Triangle

\(a:b:c=1:1:1\), \(A=60\degree\), \(B=60\degree\), \(C=60\degree\).


[Other solutions]
[Guy]
[Lob & Richmond]

\(\mathbf{2b}\) \((121)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.000000000000&{}:{}&1.366025403784&{}:{}&-1.366025403784&,\\B^\prime&{}\approx{}&-1.366025403784&{}:{}&3.732050807569&{}:{}&-1.366025403784&,\\C^\prime&{}\approx{}&-1.366025403784&{}:{}&1.366025403784&{}:{}&1.000000000000&. \end{alignedat} \]
2b (121)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-1.366025403784\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}-1.366025403784\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}-1.366025403784\overrightarrow{CI_B}. \end{aligned} \] \[ \begin{alignedat}{4} I_B&{}\approx{}&1.000000000000&{}:{}&-1.000000000000&{}:{}&1.000000000000&. \end{alignedat} \]
2b (121)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.577350269190&{}:{}&2.154700538379&{}:{}&-0.577350269190&. \end{alignedat} \]
2b (121)

Hiroyasu Kamo