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]

\(\mathbf{7c}\) \((332)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.000000000000&{}:{}&-1.366025403784&{}:{}&1.366025403784&,\\B^\prime&{}\approx{}&-1.366025403784&{}:{}&1.000000000000&{}:{}&1.366025403784&,\\C^\prime&{}\approx{}&-0.032692070451&{}:{}&-0.032692070451&{}:{}&1.065384140902&. \end{alignedat} \]
7c (332)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-1.366025403784\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-1.366025403784\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-0.032692070451\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&1.000000000000&{}:{}&1.000000000000&{}:{}&-1.000000000000&. \end{alignedat} \]
7c (332)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.108741129337&{}:{}&-0.108741129337&{}:{}&1.217482258674&. \end{alignedat} \]
7c (332)

Hiroyasu Kamo