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]

\(\mathbf{7}\) \((222)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&-0.272727272727&{}:{}&0.590318772137&{}:{}&0.682408500590&,\\B^\prime&{}\approx{}&0.660000000000&{}:{}&-0.485714285714&{}:{}&0.825714285714&,\\C^\prime&{}\approx{}&1.031250000000&{}:{}&1.116071428571&{}:{}&-1.147321428571&. \end{alignedat} \]
7 (222)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}1.818181818182\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}2.200000000000\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}3.437500000000\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&. \end{alignedat} \]
7 (222)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.407407407407&{}:{}&0.352733686067&{}:{}&0.239858906526&. \end{alignedat} \]
7 (222)

Hiroyasu Kamo