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]

\(\mathbf{0a}\) \((011)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&-1.700000000000&{}:{}&1.252319109462&{}:{}&1.447680890538&,\\B^\prime&{}\approx{}&-0.273437500000&{}:{}&0.931344696970&{}:{}&0.342092803030&,\\C^\prime&{}\approx{}&-0.311111111111&{}:{}&0.336700336700&{}:{}&0.974410774411&. \end{alignedat} \]
0a (011)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}1.542857142857\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}0.364583333333\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}0.414814814815\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&. \end{alignedat} \]
0a (011)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.528301886792&{}:{}&0.740994854202&{}:{}&0.787307032590&. \end{alignedat} \]
0a (011)

Hiroyasu Kamo