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]

\(\mathbf{0c}\) \((110)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.066203703704&{}:{}&0.424382716049&{}:{}&-0.490586419753&,\\B^\prime&{}\approx{}&0.410156250000&{}:{}&1.102982954545&{}:{}&-0.513139204545&,\\C^\prime&{}\approx{}&2.362500000000&{}:{}&2.556818181818&{}:{}&-3.919318181818&. \end{alignedat} \]
0c (110)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.325925925926\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}0.340909090909\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}1.963636363636\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&. \end{alignedat} \]
0c (110)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.945000000000&{}:{}&0.990000000000&{}:{}&-0.935000000000&. \end{alignedat} \]
0c (110)

Hiroyasu Kamo