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]

\(\mathbf{2}\) \((020)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&-0.577350269190&{}:{}&0.788675134595&{}:{}&0.788675134595&,\\B^\prime&{}\approx{}&0.018874775675&{}:{}&0.962250448649&{}:{}&0.018874775675&,\\C^\prime&{}\approx{}&0.788675134595&{}:{}&0.788675134595&{}:{}&-0.577350269190&. \end{alignedat} \]
2 (020)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}2.366025403784\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.056624327026\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}2.366025403784\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&. \end{alignedat} \]
2 (020)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.062781720295&{}:{}&0.874436559411&{}:{}&0.062781720295&. \end{alignedat} \]
2 (020)

Hiroyasu Kamo