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]

\(\mathbf{5c}\) \((312)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.000000000000&{}:{}&-5.098076211353&{}:{}&5.098076211353&,\\B^\prime&{}\approx{}&-1.098076211353&{}:{}&1.000000000000&{}:{}&1.098076211353&,\\C^\prime&{}\approx{}&-0.122008467928&{}:{}&-0.122008467928&{}:{}&1.244016935856&. \end{alignedat} \]
5c (312)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-5.098076211353\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-1.098076211353\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-0.122008467928\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&1.000000000000&{}:{}&1.000000000000&{}:{}&-1.000000000000&. \end{alignedat} \]
5c (312)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.416889446440&{}:{}&-0.138963148813&{}:{}&1.555852595253&. \end{alignedat} \]
5c (312)

Hiroyasu Kamo