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]

\(\mathbf{7c}\) \((332)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.776562500000&{}:{}&-1.432291666667&{}:{}&1.655729166667&,\\B^\prime&{}\approx{}&-1.093750000000&{}:{}&0.725378787879&{}:{}&1.368371212121&,\\C^\prime&{}\approx{}&-0.043750000000&{}:{}&-0.047348484848&{}:{}&1.091098484848&. \end{alignedat} \]
7c (332)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-1.100000000000\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-0.909090909091\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-0.036363636364\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&. \end{alignedat} \]
7c (332)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.145833333333&{}:{}&-0.152777777778&{}:{}&1.298611111111&. \end{alignedat} \]
7c (332)

Hiroyasu Kamo