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]

\(\mathbf{1c}\) \((112)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.000000000000&{}:{}&-1.366025403784&{}:{}&1.366025403784&,\\B^\prime&{}\approx{}&-1.366025403784&{}:{}&1.000000000000&{}:{}&1.366025403784&,\\C^\prime&{}\approx{}&-1.366025403784&{}:{}&-1.366025403784&{}:{}&3.732050807569&. \end{alignedat} \]
1c (112)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-1.366025403784\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-1.366025403784\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-1.366025403784\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&1.000000000000&{}:{}&1.000000000000&{}:{}&-1.000000000000&. \end{alignedat} \]
1c (112)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.577350269190&{}:{}&-0.577350269190&{}:{}&2.154700538379&. \end{alignedat} \]
1c (112)

Hiroyasu Kamo