Derousseau's Generalization of the Malfatti circles

Martin's solution

Problem 4331 (proposed by A. Martin) I. Solution by the Proposer, Mathematical Questions with their Solutions, from the “Educational Times.”.

\(a:b:c=231:250:289\).


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

\(\mathbf{4b}\) \((301)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.255208333333&{}:{}&-4.774305555556&{}:{}&5.519097222222&,\\B^\prime&{}\approx{}&0.373333333333&{}:{}&0.159595959596&{}:{}&0.467070707071&,\\C^\prime&{}\approx{}&0.064814814815&{}:{}&-0.070145903479&{}:{}&1.005331088664&. \end{alignedat} \]
4b (301)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}5.156250000000\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}0.436363636364\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}0.075757575758\overrightarrow{CI_B}. \end{aligned} \] \[ \begin{alignedat}{4} I_B&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&. \end{alignedat} \]
4b (301)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.215799614644&{}:{}&-0.176621708414&{}:{}&0.960822093770&. \end{alignedat} \]
4b (301)

Hiroyasu Kamo