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]

\(\mathbf{4}\) \((200)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.976569900005&{}:{}&0.010413377775&{}:{}&0.013016722219&,\\B^\prime&{}\approx{}&0.686508254706&{}:{}&-0.830688679216&{}:{}&1.144180424510&,\\C^\prime&{}\approx{}&1.167552905630&{}:{}&1.556737207507&{}:{}&-1.724290113137&. \end{alignedat} \]
4 (200)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.031240133326\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}2.746033018824\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}4.670211622521\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.250000000000&{}:{}&0.333333333333&{}:{}&0.416666666667&. \end{alignedat} \]
4 (200)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.920624914364&{}:{}&0.033250813799&{}:{}&0.046124271837&. \end{alignedat} \]
4 (200)

Hiroyasu Kamo