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]

\(\mathbf{1a}\) \((013)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.089316397477&{}:{}&0.455341801261&{}:{}&0.455341801261&,\\B^\prime&{}\approx{}&-0.098076211353&{}:{}&1.000000000000&{}:{}&0.098076211353&,\\C^\prime&{}\approx{}&-4.098076211353&{}:{}&4.098076211353&{}:{}&1.000000000000&. \end{alignedat} \]
1a (013)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.455341801261\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}0.098076211353\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}4.098076211353\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-1.000000000000&{}:{}&1.000000000000&{}:{}&1.000000000000&. \end{alignedat} \]
1a (013)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.251504769166&{}:{}&0.938628576875&{}:{}&0.312876192292&. \end{alignedat} \]
1a (013)

Hiroyasu Kamo