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]

\(\mathbf{1b}\) \((103)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.982445987654&{}:{}&-0.112525720165&{}:{}&0.130079732510&,\\B^\prime&{}\approx{}&0.440000000000&{}:{}&0.009523809524&{}:{}&0.550476190476&,\\C^\prime&{}\approx{}&2.750000000000&{}:{}&-2.976190476190&{}:{}&1.226190476190&. \end{alignedat} \]
1b (103)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.121527777778\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}0.514285714286\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}3.214285714286\overrightarrow{CI_B}. \end{aligned} \] \[ \begin{alignedat}{4} I_B&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&. \end{alignedat} \]
1b (103)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.893401015228&{}:{}&-0.296108291032&{}:{}&0.402707275804&. \end{alignedat} \]
1b (103)

Hiroyasu Kamo