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]

\(\mathbf{0b}\) \((101)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.000000000000&{}:{}&-0.366025403784&{}:{}&0.366025403784&,\\B^\prime&{}\approx{}&1.699358737118&{}:{}&-2.398717474236&{}:{}&1.699358737118&,\\C^\prime&{}\approx{}&0.366025403784&{}:{}&-0.366025403784&{}:{}&1.000000000000&. \end{alignedat} \]
0b (101)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.366025403784\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}1.699358737118\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}0.366025403784\overrightarrow{CI_B}. \end{aligned} \] \[ \begin{alignedat}{4} I_B&{}\approx{}&1.000000000000&{}:{}&-1.000000000000&{}:{}&1.000000000000&. \end{alignedat} \]
0b (101)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.836013856610&{}:{}&-0.672027713219&{}:{}&0.836013856610&. \end{alignedat} \]
0b (101)

Hiroyasu Kamo