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]

\(\mathbf{4a}\) \((211)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&2.298631302823&{}:{}&-0.577169467921&{}:{}&-0.721461834902&,\\B^\prime&{}\approx{}&1.683049043273&{}:{}&2.122032695515&{}:{}&-2.805081738788&,\\C^\prime&{}\approx{}&0.989614308448&{}:{}&-1.319485744597&{}:{}&1.329871436149&. \end{alignedat} \]
4a (211)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-0.865754201882\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-3.366098086545\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-1.979228616896\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&. \end{alignedat} \]
4a (211)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&1.711705482154&{}:{}&-0.211869851448&{}:{}&-0.499835630705&. \end{alignedat} \]
4a (211)

Hiroyasu Kamo