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]

\(\mathbf{3}\) \((022)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&-0.718181818182&{}:{}&0.796930342385&{}:{}&0.921251475797&,\\B^\prime&{}\approx{}&0.122222222222&{}:{}&0.724867724868&{}:{}&0.152910052910&,\\C^\prime&{}\approx{}&0.275000000000&{}:{}&0.297619047619&{}:{}&0.427380952381&. \end{alignedat} \]
3 (022)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}2.454545454545\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.407407407407\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}0.916666666667\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&. \end{alignedat} \]
3 (022)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.032835820896&{}:{}&0.575692963753&{}:{}&0.391471215352&. \end{alignedat} \]
3 (022)

Hiroyasu Kamo