Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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

\(\mathbf{4c}\) \((310)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&4.314755078384&{}:{}&13.259020313537&{}:{}&-16.573775391922&,\\B^\prime&{}\approx{}&0.113130530351&{}:{}&1.075420353567&{}:{}&-0.188550883918&,\\C^\prime&{}\approx{}&0.387704095259&{}:{}&0.516938793679&{}:{}&0.095357111061&. \end{alignedat} \]
4c (310)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}6.629510156769\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}0.075420353567\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}0.258469396840\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&1.500000000000&{}:{}&2.000000000000&{}:{}&-2.500000000000&. \end{alignedat} \]
4c (310)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.346053821241&{}:{}&1.098641704032&{}:{}&-0.444695525273&. \end{alignedat} \]
4c (310)

Hiroyasu Kamo