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]

\(\mathbf{5b}\) \((303)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.905208333333&{}:{}&-0.607638888889&{}:{}&0.702430555556&,\\B^\prime&{}\approx{}&0.325925925926&{}:{}&0.266313932981&{}:{}&0.407760141093&,\\C^\prime&{}\approx{}&0.183333333333&{}:{}&-0.198412698413&{}:{}&1.015079365079&. \end{alignedat} \]
5b (303)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.656250000000\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}0.380952380952\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}0.214285714286\overrightarrow{CI_B}. \end{aligned} \] \[ \begin{alignedat}{4} I_B&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&. \end{alignedat} \]
5b (303)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.404597701149&{}:{}&-0.134099616858&{}:{}&0.729501915709&. \end{alignedat} \]
5b (303)

Hiroyasu Kamo