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]

\(\mathbf{5a}\) \((213)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.254545454545&{}:{}&-0.118063754427&{}:{}&-0.136481700118&,\\B^\prime&{}\approx{}&1.289062500000&{}:{}&1.323660714286&{}:{}&-1.612723214286&,\\C^\prime&{}\approx{}&3.300000000000&{}:{}&-3.571428571429&{}:{}&1.271428571429&. \end{alignedat} \]
5a (213)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-0.145454545455\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-1.718750000000\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-4.400000000000\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&. \end{alignedat} \]
5a (213)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&1.571428571429&{}:{}&-0.108843537415&{}:{}&-0.462585034014&. \end{alignedat} \]
5a (213)

Hiroyasu Kamo