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]

\(\mathbf{7a}\) \((233)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.032507527261&{}:{}&-0.014447789894&{}:{}&-0.018059737367&,\\B^\prime&{}\approx{}&1.979228616896&{}:{}&2.319485744597&{}:{}&-3.298714361493&,\\C^\prime&{}\approx{}&0.841524521636&{}:{}&-1.122032695515&{}:{}&1.280508173879&. \end{alignedat} \]
7a (233)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-0.021671684841\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-3.958457233792\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-1.683049043273\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&. \end{alignedat} \]
7a (233)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&1.107885529448&{}:{}&-0.052379304788&{}:{}&-0.055506224660&. \end{alignedat} \]
7a (233)

Hiroyasu Kamo