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]

\(\mathbf{3b}\) \((123)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.537910698151&{}:{}&2.151642792606&{}:{}&-2.689553490757&,\\B^\prime&{}\approx{}&-0.174285434966&{}:{}&1.464761159908&{}:{}&-0.290475724943&,\\C^\prime&{}\approx{}&-2.389140284973&{}:{}&3.185520379965&{}:{}&0.203619905009&. \end{alignedat} \]
3b (123)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-2.151642792606\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}-0.232380579954\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}-3.185520379965\overrightarrow{CI_B}. \end{aligned} \] \[ \begin{alignedat}{4} I_B&{}\approx{}&0.750000000000&{}:{}&-1.000000000000&{}:{}&1.250000000000&. \end{alignedat} \]
3b (123)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.122948098990&{}:{}&2.043805322007&{}:{}&-0.920857223017&. \end{alignedat} \]
3b (123)

Hiroyasu Kamo