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]

\(\mathbf{2a}\) \((031)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.140909090909&{}:{}&0.398465171192&{}:{}&0.460625737898&,\\B^\prime&{}\approx{}&-3.437500000000&{}:{}&0.136904761905&{}:{}&4.300595238095&,\\C^\prime&{}\approx{}&-0.061111111111&{}:{}&0.066137566138&{}:{}&0.994973544974&. \end{alignedat} \]
2a (031)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.490909090909\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}4.583333333333\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}0.081481481481\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&. \end{alignedat} \]
2a (031)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.157142857143&{}:{}&0.220408163265&{}:{}&0.936734693878&. \end{alignedat} \]
2a (031)

Hiroyasu Kamo