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]

\(\mathbf{0b}\) \((101)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.950347222222&{}:{}&-0.318287037037&{}:{}&0.367939814815&,\\B^\prime&{}\approx{}&1.400000000000&{}:{}&-2.151515151515&{}:{}&1.751515151515&,\\C^\prime&{}\approx{}&0.350000000000&{}:{}&-0.378787878788&{}:{}&1.028787878788&. \end{alignedat} \]
0b (101)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.343750000000\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}1.636363636364\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}0.409090909091\overrightarrow{CI_B}. \end{aligned} \] \[ \begin{alignedat}{4} I_B&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&. \end{alignedat} \]
0b (101)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.772413793103&{}:{}&-0.632183908046&{}:{}&0.859770114943&. \end{alignedat} \]
0b (101)

Hiroyasu Kamo