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]

\(\mathbf{6c}\) \((330)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.078993055556&{}:{}&0.506365740741&{}:{}&-0.585358796296&,\\B^\prime&{}\approx{}&0.343750000000&{}:{}&1.086309523810&{}:{}&-0.430059523810&,\\C^\prime&{}\approx{}&0.343750000000&{}:{}&0.372023809524&{}:{}&0.284226190476&. \end{alignedat} \]
6c (330)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.388888888889\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}0.285714285714\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}0.285714285714\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&. \end{alignedat} \]
6c (330)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.562500000000&{}:{}&0.662878787879&{}:{}&-0.225378787879&. \end{alignedat} \]
6c (330)

Hiroyasu Kamo