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]

\(\mathbf{6}\) \((220)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.600000000000&{}:{}&0.185528756957&{}:{}&0.214471243043&,\\B^\prime&{}\approx{}&0.233333333333&{}:{}&0.474747474747&{}:{}&0.291919191919&,\\C^\prime&{}\approx{}&1.181250000000&{}:{}&1.278409090909&{}:{}&-1.459659090909&. \end{alignedat} \]
6 (220)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.571428571429\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.777777777778\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}3.937500000000\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&. \end{alignedat} \]
6 (220)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.517808219178&{}:{}&0.448318804483&{}:{}&0.033872976339&. \end{alignedat} \]
6 (220)

Hiroyasu Kamo