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]

\(\mathbf{2}\) \((020)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&-0.500000000000&{}:{}&0.695732838590&{}:{}&0.804267161410&,\\B^\prime&{}\approx{}&0.015555555556&{}:{}&0.964983164983&{}:{}&0.019461279461&,\\C^\prime&{}\approx{}&0.875000000000&{}:{}&0.946969696970&{}:{}&-0.821969696970&. \end{alignedat} \]
2 (020)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}2.142857142857\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.051851851852\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}2.916666666667\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&. \end{alignedat} \]
2 (020)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.050359712230&{}:{}&0.882930019621&{}:{}&0.066710268149&. \end{alignedat} \]
2 (020)

Hiroyasu Kamo