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]

\(\mathbf{6b}\) \((321)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.947916666667&{}:{}&6.076388888889&{}:{}&-7.024305555556&,\\B^\prime&{}\approx{}&-0.073333333333&{}:{}&1.165079365079&{}:{}&-0.091746031746&,\\C^\prime&{}\approx{}&-0.814814814815&{}:{}&0.881834215168&{}:{}&0.932980599647&. \end{alignedat} \]
6b (321)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-6.562500000000\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}-0.085714285714\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}-0.952380952381\overrightarrow{CI_B}. \end{aligned} \] \[ \begin{alignedat}{4} I_B&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&. \end{alignedat} \]
6b (321)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.259587020649&{}:{}&1.376597836775&{}:{}&-0.117010816126&. \end{alignedat} \]
6b (321)

Hiroyasu Kamo