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]

\(\mathbf{4}\) \((200)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.968181818182&{}:{}&0.014757969303&{}:{}&0.017060212515&,\\B^\prime&{}\approx{}&0.733333333333&{}:{}&-0.650793650794&{}:{}&0.917460317460&,\\C^\prime&{}\approx{}&0.928125000000&{}:{}&1.004464285714&{}:{}&-0.932589285714&. \end{alignedat} \]
4 (200)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.045454545455\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}2.444444444444\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}3.093750000000\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&. \end{alignedat} \]
4 (200)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.893233082707&{}:{}&0.048335123523&{}:{}&0.058431793770&. \end{alignedat} \]
4 (200)

Hiroyasu Kamo