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]

\(\mathbf{0}\) \((000)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.522727272727&{}:{}&0.221369539551&{}:{}&0.255903187721&,\\B^\prime&{}\approx{}&0.195555555556&{}:{}&0.559788359788&{}:{}&0.244656084656&,\\C^\prime&{}\approx{}&0.171875000000&{}:{}&0.186011904762&{}:{}&0.642113095238&. \end{alignedat} \]
0 (000)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.681818181818\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.651851851852\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}0.572916666667\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&. \end{alignedat} \]
0 (000)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.292358803987&{}:{}&0.320360702421&{}:{}&0.387280493593&. \end{alignedat} \]
0 (000)

Hiroyasu Kamo