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]

\(\mathbf{4a}\) \((211)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&3.000000000000&{}:{}&-0.927643784787&{}:{}&-1.072356215213&,\\B^\prime&{}\approx{}&1.476562500000&{}:{}&1.370738636364&{}:{}&-1.847301136364&,\\C^\prime&{}\approx{}&1.166666666667&{}:{}&-1.262626262626&{}:{}&1.095959595960&. \end{alignedat} \]
4a (211)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-1.142857142857\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-1.968750000000\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-1.555555555556\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&. \end{alignedat} \]
4a (211)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&1.948453608247&{}:{}&-0.374882849110&{}:{}&-0.573570759138&. \end{alignedat} \]
4a (211)

Hiroyasu Kamo