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]

\(\mathbf{1a}\) \((013)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.010385691552&{}:{}&0.439828581532&{}:{}&0.549785726915&,\\B^\prime&{}\approx{}&-0.065015054523&{}:{}&0.956656630318&{}:{}&0.108358424204&,\\C^\prime&{}\approx{}&-1.298631302823&{}:{}&1.731508403764&{}:{}&0.567122899059&. \end{alignedat} \]
1a (013)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.659742872299\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}0.130030109045\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}2.597262605646\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&. \end{alignedat} \]
1a (013)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.178482548008&{}:{}&0.838915857229&{}:{}&0.339566690779&. \end{alignedat} \]
1a (013)

Hiroyasu Kamo