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]

\(\mathbf{1b}\) \((103)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.962089301849&{}:{}&-0.151642792606&{}:{}&0.189553490757&,\\B^\prime&{}\approx{}&0.424285434966&{}:{}&-0.131427826575&{}:{}&0.707142391609&,\\C^\prime&{}\approx{}&1.889140284973&{}:{}&-2.518853713298&{}:{}&1.629713428324&. \end{alignedat} \]
1b (103)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.151642792606\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}0.565713913288\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}2.518853713298\overrightarrow{CI_B}. \end{aligned} \] \[ \begin{alignedat}{4} I_B&{}\approx{}&0.750000000000&{}:{}&-1.000000000000&{}:{}&1.250000000000&. \end{alignedat} \]
1b (103)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.842941130372&{}:{}&-0.405666854564&{}:{}&0.562725724193&. \end{alignedat} \]
1b (103)

Hiroyasu Kamo