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]

\(\mathbf{1c}\) \((112)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.789351851852&{}:{}&-1.350308641975&{}:{}&1.560956790123&,\\B^\prime&{}\approx{}&-1.160156250000&{}:{}&0.708705357143&{}:{}&1.451450892857&,\\C^\prime&{}\approx{}&-2.062500000000&{}:{}&-2.232142857143&{}:{}&5.294642857143&. \end{alignedat} \]
1c (112)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-1.037037037037\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-0.964285714286\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-1.714285714286\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&. \end{alignedat} \]
1c (112)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.700471698113&{}:{}&-0.825471698113&{}:{}&2.525943396226&. \end{alignedat} \]
1c (112)

Hiroyasu Kamo