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]

\(\mathbf{7a}\) \((233)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.050000000000&{}:{}&-0.023191094620&{}:{}&-0.026808905380&,\\B^\prime&{}\approx{}&1.640625000000&{}:{}&1.411931818182&{}:{}&-2.052556818182&,\\C^\prime&{}\approx{}&1.050000000000&{}:{}&-1.136363636364&{}:{}&1.086363636364&. \end{alignedat} \]
7a (233)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-0.028571428571\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-2.187500000000\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-1.400000000000\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&. \end{alignedat} \]
7a (233)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&1.166666666667&{}:{}&-0.080808080808&{}:{}&-0.085858585859&. \end{alignedat} \]
7a (233)

Hiroyasu Kamo