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]

\(\mathbf{3a}\) \((033)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.217492472739&{}:{}&0.347781123227&{}:{}&0.434726404034&,\\B^\prime&{}\approx{}&-0.479228616896&{}:{}&0.680514255403&{}:{}&0.798714361493&,\\C^\prime&{}\approx{}&-0.091524521636&{}:{}&0.122032695515&{}:{}&0.969491826121&. \end{alignedat} \]
3a (033)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.521671684841\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}0.958457233792\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}0.183049043273\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&. \end{alignedat} \]
3a (033)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.067617961931&{}:{}&0.317822560974&{}:{}&0.749795400956&. \end{alignedat} \]
3a (033)

Hiroyasu Kamo