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]

\(\mathbf{7}\) \((222)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&,\\B^\prime&{}\approx{}&0.500000000000&{}:{}&-0.505952380952&{}:{}&1.005952380952&,\\C^\prime&{}\approx{}&2.000000000000&{}:{}&2.976190476190&{}:{}&-3.976190476190&. \end{alignedat} \]
7 (222)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}1.000000000000\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}2.250000000000\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}9.000000000000\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&. \end{alignedat} \]
7 (222)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.571428571429&{}:{}&0.340136054422&{}:{}&0.088435374150&. \end{alignedat} \]
7 (222)

Hiroyasu Kamo