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]

\(\mathbf{7a}\) \((233)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.014820022545&{}:{}&-0.006175009394&{}:{}&-0.008645013151&,\\B^\prime&{}\approx{}&4.017825533194&{}:{}&6.357100710925&{}:{}&-9.374926244119&,\\C^\prime&{}\approx{}&0.622334958066&{}:{}&-1.037224930110&{}:{}&1.414889972044&. \end{alignedat} \]
7a (233)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-0.011115016909\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-12.053476599582\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-1.867004874198\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&. \end{alignedat} \]
7a (233)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&1.047294131480&{}:{}&-0.023696387043&{}:{}&-0.023597744437&. \end{alignedat} \]
7a (233)

Hiroyasu Kamo