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]

\(\mathbf{3c}\) \((132)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.834490740741&{}:{}&-1.060956790123&{}:{}&1.226466049383&,\\B^\prime&{}\approx{}&-5.906250000000&{}:{}&-0.482954545455&{}:{}&7.389204545455&,\\C^\prime&{}\approx{}&-0.164062500000&{}:{}&-0.177556818182&{}:{}&1.341619318182&. \end{alignedat} \]
3c (132)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-0.814814814815\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-4.909090909091\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-0.136363636364\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&. \end{alignedat} \]
3c (132)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.200211864407&{}:{}&-0.582627118644&{}:{}&1.782838983051&. \end{alignedat} \]
3c (132)

Hiroyasu Kamo