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]

\(\mathbf{5}\) \((202)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.750000000000&{}:{}&0.115955473098&{}:{}&0.134044526902&,\\B^\prime&{}\approx{}&0.840000000000&{}:{}&-0.890909090909&{}:{}&1.050909090909&,\\C^\prime&{}\approx{}&0.328125000000&{}:{}&0.355113636364&{}:{}&0.316761363636&. \end{alignedat} \]
5 (202)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.357142857143\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}2.800000000000\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}1.093750000000\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&. \end{alignedat} \]
5 (202)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.608695652174&{}:{}&0.032938076416&{}:{}&0.358366271410&. \end{alignedat} \]
5 (202)

Hiroyasu Kamo