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]

\(\mathbf{3}\) \((022)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&-0.476569900005&{}:{}&0.656253288891&{}:{}&0.820316611114&,\\B^\prime&{}\approx{}&0.063491745294&{}:{}&0.830688679216&{}:{}&0.105819575490&,\\C^\prime&{}\approx{}&0.332447094370&{}:{}&0.443262792493&{}:{}&0.224290113137&. \end{alignedat} \]
3 (022)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}1.968759866674\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.253966981176\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}1.329788377479\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.250000000000&{}:{}&0.333333333333&{}:{}&0.416666666667&. \end{alignedat} \]
3 (022)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.027248234267&{}:{}&0.670604227497&{}:{}&0.302147538235&. \end{alignedat} \]
3 (022)

Hiroyasu Kamo