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]

\(\mathbf{0}\) \((000)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.577350269190&{}:{}&0.211324865405&{}:{}&0.211324865405&,\\B^\prime&{}\approx{}&0.211324865405&{}:{}&0.577350269190&{}:{}&0.211324865405&,\\C^\prime&{}\approx{}&0.211324865405&{}:{}&0.211324865405&{}:{}&0.577350269190&. \end{alignedat} \]
0 (000)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.633974596216\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.633974596216\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}0.633974596216\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&. \end{alignedat} \]
0 (000)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&. \end{alignedat} \]
0 (000)

Hiroyasu Kamo