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]

\(\mathbf{1a}\) \((013)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.045454545455&{}:{}&0.442739079103&{}:{}&0.511806375443&,\\B^\prime&{}\approx{}&-0.085937500000&{}:{}&0.978422619048&{}:{}&0.107514880952&,\\C^\prime&{}\approx{}&-2.444444444444&{}:{}&2.645502645503&{}:{}&0.798941798942&. \end{alignedat} \]
1a (013)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.545454545455\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}0.114583333333\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}3.259259259259\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&. \end{alignedat} \]
1a (013)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.227979274611&{}:{}&0.888230940044&{}:{}&0.339748334567&. \end{alignedat} \]
1a (013)

Hiroyasu Kamo