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]

\(\mathbf{5a}\) \((213)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.239614308448&{}:{}&-0.106495248199&{}:{}&-0.133119060249&,\\B^\prime&{}\approx{}&1.565015054523&{}:{}&2.043343369682&{}:{}&-2.608358424204&,\\C^\prime&{}\approx{}&2.048631302823&{}:{}&-2.731508403764&{}:{}&1.682877100941&. \end{alignedat} \]
5a (213)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-0.159742872299\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-3.130030109045\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-4.097262605646\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&. \end{alignedat} \]
5a (213)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&1.513520868154&{}:{}&-0.071557185962&{}:{}&-0.441963682191&. \end{alignedat} \]
5a (213)

Hiroyasu Kamo