Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


[Other solutions]
[Guy]
[Lob & Richmond]

\(\mathbf{5a}\) \((213)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.271420142185&{}:{}&-0.113091725910&{}:{}&-0.158328416275&,\\B^\prime&{}\approx{}&3.055575084545&{}:{}&5.074100112726&{}:{}&-7.129675197271&,\\C^\prime&{}\approx{}&1.199685227256&{}:{}&-1.999475378759&{}:{}&1.799790151504&. \end{alignedat} \]
5a (213)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-0.203565106639\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-9.166725253634\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-3.599055681767\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&. \end{alignedat} \]
5a (213)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&1.507063777257&{}:{}&-0.034099271151&{}:{}&-0.472964506106&. \end{alignedat} \]
5a (213)

Hiroyasu Kamo