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]

\(\mathbf{6a}\) \((231)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.159090909091&{}:{}&-0.073789846517&{}:{}&-0.085301062574&,\\B^\prime&{}\approx{}&4.640625000000&{}:{}&2.165178571429&{}:{}&-5.805803571429&,\\C^\prime&{}\approx{}&0.916666666667&{}:{}&-0.992063492063&{}:{}&1.075396825397&. \end{alignedat} \]
6a (231)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-0.090909090909\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-6.187500000000\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-1.222222222222\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&. \end{alignedat} \]
6a (231)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&1.362385321101&{}:{}&-0.262123197903&{}:{}&-0.100262123198&. \end{alignedat} \]
6a (231)

Hiroyasu Kamo