Derousseau's Generalization of the Malfatti circles

\(A=45\degree\), \(B=45\degree\), \(C=90\degree\).


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

\(\mathbf{0}\) \((000)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.500000000000&{}:{}&0.207106781187&{}:{}&0.292893218813&,\\B^\prime&{}\approx{}&0.207106781187&{}:{}&0.500000000000&{}:{}&0.292893218813&,\\C^\prime&{}\approx{}&0.148846697643&{}:{}&0.148846697643&{}:{}&0.702306604714&. \end{alignedat} \]
0 (000)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.707106781187\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.707106781187\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}0.508194413807\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.292893218813&{}:{}&0.292893218813&{}:{}&0.414213562373&. \end{alignedat} \]
0 (000)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.278257463541&{}:{}&0.278257463541&{}:{}&0.443485072918&. \end{alignedat} \]
0 (000)

Hiroyasu Kamo