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]

\(\mathbf{0}\) \((000)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.436000484963&{}:{}&0.250666451128&{}:{}&0.313333063909&,\\B^\prime&{}\approx{}&0.166223547185&{}:{}&0.556737207507&{}:{}&0.277039245308&,\\C^\prime&{}\approx{}&0.126983490588&{}:{}&0.169311320784&{}:{}&0.703705188628&. \end{alignedat} \]
0 (000)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.751999353383\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.664894188740\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}0.507933962352\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.250000000000&{}:{}&0.333333333333&{}:{}&0.416666666667&. \end{alignedat} \]
0 (000)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.233970657965&{}:{}&0.320895389436&{}:{}&0.445133952599&. \end{alignedat} \]
0 (000)

Hiroyasu Kamo