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]

\(\mathbf{1}\) \((002)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&-0.350000000000&{}:{}&0.626159554731&{}:{}&0.723840445269&,\\B^\prime&{}\approx{}&0.622222222222&{}:{}&-0.400673400673&{}:{}&0.778451178451&,\\C^\prime&{}\approx{}&0.021875000000&{}:{}&0.023674242424&{}:{}&0.954450757576&. \end{alignedat} \]
1 (002)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}1.928571428571\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}2.074074074074\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}0.072916666667\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&. \end{alignedat} \]
1 (002)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.071337579618&{}:{}&0.078170237406&{}:{}&0.850492182976&. \end{alignedat} \]
1 (002)

Hiroyasu Kamo