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]

\(\mathbf{4c}\) \((310)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.000000000000&{}:{}&4.098076211353&{}:{}&-4.098076211353&,\\B^\prime&{}\approx{}&0.098076211353&{}:{}&1.000000000000&{}:{}&-0.098076211353&,\\C^\prime&{}\approx{}&0.455341801261&{}:{}&0.455341801261&{}:{}&0.089316397477&. \end{alignedat} \]
4c (310)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}4.098076211353\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}0.098076211353\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}0.455341801261\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&1.000000000000&{}:{}&1.000000000000&{}:{}&-1.000000000000&. \end{alignedat} \]
4c (310)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.312876192292&{}:{}&0.938628576875&{}:{}&-0.251504769166&. \end{alignedat} \]
4c (310)

Hiroyasu Kamo