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]

\(\mathbf{7b}\) \((323)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.297916666667&{}:{}&1.909722222222&{}:{}&-2.207638888889&,\\B^\prime&{}\approx{}&-0.025925925926&{}:{}&1.058361391695&{}:{}&-0.032435465769&,\\C^\prime&{}\approx{}&-0.933333333333&{}:{}&1.010101010101&{}:{}&0.923232323232&. \end{alignedat} \]
7b (323)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-2.062500000000\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}-0.030303030303\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}-1.090909090909\overrightarrow{CI_B}. \end{aligned} \] \[ \begin{alignedat}{4} I_B&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&. \end{alignedat} \]
7b (323)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.091056910569&{}:{}&1.192411924119&{}:{}&-0.101355013550&. \end{alignedat} \]
7b (323)

Hiroyasu Kamo