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]

\(\mathbf{3b}\) \((123)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.220679012346&{}:{}&1.414609053498&{}:{}&-1.635288065844&,\\B^\prime&{}\approx{}&-0.140000000000&{}:{}&1.315151515152&{}:{}&-0.175151515152&,\\C^\prime&{}\approx{}&-3.500000000000&{}:{}&3.787878787879&{}:{}&0.712121212121&. \end{alignedat} \]
3b (123)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-1.527777777778\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}-0.163636363636\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}-4.090909090909\overrightarrow{CI_B}. \end{aligned} \] \[ \begin{alignedat}{4} I_B&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&. \end{alignedat} \]
3b (123)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.130841121495&{}:{}&1.713395638629&{}:{}&-0.582554517134&. \end{alignedat} \]
3b (123)

Hiroyasu Kamo