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]

\(\mathbf{0a}\) \((011)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&-0.599580303007&{}:{}&0.666491792920&{}:{}&0.933088510088&,\\B^\prime&{}\approx{}&-0.111674790330&{}:{}&0.851100279560&{}:{}&0.260574510770&,\\C^\prime&{}\approx{}&-0.203565106639&{}:{}&0.339275177731&{}:{}&0.864289928907&. \end{alignedat} \]
0a (011)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}1.199685227256\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}0.335024370989\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}0.610695319916\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&. \end{alignedat} \]
0a (011)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.252078775602&{}:{}&0.627345134507&{}:{}&0.624733641096&. \end{alignedat} \]
0a (011)

Hiroyasu Kamo