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]

\(\mathbf{3a}\) \((033)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.250000000000&{}:{}&0.347866419295&{}:{}&0.402133580705&,\\B^\prime&{}\approx{}&-0.437500000000&{}:{}&0.890151515152&{}:{}&0.547348484848&,\\C^\prime&{}\approx{}&-0.194444444444&{}:{}&0.210437710438&{}:{}&0.984006734007&. \end{alignedat} \]
3a (033)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.428571428571\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}0.583333333333\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}0.259259259259\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&. \end{alignedat} \]
3a (033)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.112903225806&{}:{}&0.439882697947&{}:{}&0.673020527859&. \end{alignedat} \]
3a (033)

Hiroyasu Kamo