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]

\(\mathbf{6a}\) \((231)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.244016935856&{}:{}&-0.122008467928&{}:{}&-0.122008467928&,\\B^\prime&{}\approx{}&5.098076211353&{}:{}&1.000000000000&{}:{}&-5.098076211353&,\\C^\prime&{}\approx{}&1.098076211353&{}:{}&-1.098076211353&{}:{}&1.000000000000&. \end{alignedat} \]
6a (231)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-0.122008467928\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-5.098076211353\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-1.098076211353\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-1.000000000000&{}:{}&1.000000000000&{}:{}&1.000000000000&. \end{alignedat} \]
6a (231)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&1.555852595253&{}:{}&-0.416889446440&{}:{}&-0.138963148813&. \end{alignedat} \]
6a (231)

Hiroyasu Kamo