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]

\(\mathbf{3b}\) \((123)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.000000000000&{}:{}&1.098076211353&{}:{}&-1.098076211353&,\\B^\prime&{}\approx{}&-0.122008467928&{}:{}&1.244016935856&{}:{}&-0.122008467928&,\\C^\prime&{}\approx{}&-5.098076211353&{}:{}&5.098076211353&{}:{}&1.000000000000&. \end{alignedat} \]
3b (123)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-1.098076211353\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}-0.122008467928\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}-5.098076211353\overrightarrow{CI_B}. \end{aligned} \] \[ \begin{alignedat}{4} I_B&{}\approx{}&1.000000000000&{}:{}&-1.000000000000&{}:{}&1.000000000000&. \end{alignedat} \]
3b (123)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.138963148813&{}:{}&1.555852595253&{}:{}&-0.416889446440&. \end{alignedat} \]
3b (123)

Hiroyasu Kamo