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]

\(\mathbf{4a}\) \((211)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&3.732050807569&{}:{}&-1.366025403784&{}:{}&-1.366025403784&,\\B^\prime&{}\approx{}&1.366025403784&{}:{}&1.000000000000&{}:{}&-1.366025403784&,\\C^\prime&{}\approx{}&1.366025403784&{}:{}&-1.366025403784&{}:{}&1.000000000000&. \end{alignedat} \]
4a (211)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-1.366025403784\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-1.366025403784\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-1.366025403784\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-1.000000000000&{}:{}&1.000000000000&{}:{}&1.000000000000&. \end{alignedat} \]
4a (211)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&2.154700538379&{}:{}&-0.577350269190&{}:{}&-0.577350269190&. \end{alignedat} \]
4a (211)

Hiroyasu Kamo