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]

\(\mathbf{5c}\) \((312)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&-0.137500000000&{}:{}&-7.291666666667&{}:{}&8.429166666667&,\\B^\prime&{}\approx{}&-0.859375000000&{}:{}&0.784226190476&{}:{}&1.075148809524&,\\C^\prime&{}\approx{}&-0.137500000000&{}:{}&-0.148809523810&{}:{}&1.286309523810&. \end{alignedat} \]
5c (312)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-5.600000000000\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-0.714285714286\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-0.114285714286\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&. \end{alignedat} \]
5c (312)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.458333333333&{}:{}&-0.194444444444&{}:{}&1.652777777778&. \end{alignedat} \]
5c (312)

Hiroyasu Kamo