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]

\(\mathbf{4c}\) \((310)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.993055555556&{}:{}&6.365740740741&{}:{}&-7.358796296296&,\\B^\prime&{}\approx{}&0.109375000000&{}:{}&1.027462121212&{}:{}&-0.136837121212&,\\C^\prime&{}\approx{}&0.437500000000&{}:{}&0.473484848485&{}:{}&0.089015151515&. \end{alignedat} \]
4c (310)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}4.888888888889\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}0.090909090909\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}0.363636363636\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&. \end{alignedat} \]
4c (310)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.342391304348&{}:{}&0.996376811594&{}:{}&-0.338768115942&. \end{alignedat} \]
4c (310)

Hiroyasu Kamo