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]

\(\mathbf{1b}\) \((103)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.922629435690&{}:{}&-0.193426410776&{}:{}&0.270796975086&,\\B^\prime&{}\approx{}&0.415583153339&{}:{}&-0.385277177798&{}:{}&0.969694024458&,\\C^\prime&{}\approx{}&1.058482258555&{}:{}&-1.764137097591&{}:{}&1.705654839036&. \end{alignedat} \]
1b (103)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.193426410776\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}0.692638588899\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}1.764137097591\overrightarrow{CI_B}. \end{aligned} \] \[ \begin{alignedat}{4} I_B&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&. \end{alignedat} \]
1b (103)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.758075783567&{}:{}&-0.526856043671&{}:{}&0.768780260103&. \end{alignedat} \]
1b (103)

Hiroyasu Kamo