Derousseau's Generalization of the Malfatti circles

Ajima's example

不朽算法 十四 (Fukyū Sanpō §14)

\(a=252\),  \(b=375\),  \(c=507\).  \(a:b:c=84:125:169\).


[Other solutions]
[Guy]
[Lob & Richmond]

\(\mathbf{0}\) \((000)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.377777777778&{}:{}&0.264550264550&{}:{}&0.357671957672&,\\B^\prime&{}\approx{}&0.156250000000&{}:{}&0.529389880952&{}:{}&0.314360119048&,\\C^\prime&{}\approx{}&0.100000000000&{}:{}&0.148809523810&{}:{}&0.751190476190&. \end{alignedat} \]
0 (000)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.800000000000\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.703125000000\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}0.450000000000\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&. \end{alignedat} \]
0 (000)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.200000000000&{}:{}&0.304761904762&{}:{}&0.495238095238&. \end{alignedat} \]
0 (000)

Hiroyasu Kamo