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]

\(\mathbf{2c}\) \((130)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.021064814815&{}:{}&0.135030864198&{}:{}&-0.156095679012&,\\B^\prime&{}\approx{}&5.156250000000&{}:{}&2.294642857143&{}:{}&-6.450892857143&,\\C^\prime&{}\approx{}&0.464062500000&{}:{}&0.502232142857&{}:{}&0.033705357143&. \end{alignedat} \]
2c (130)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.103703703704\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}4.285714285714\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}0.385714285714\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&. \end{alignedat} \]
2c (130)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.976973684211&{}:{}&0.414473684211&{}:{}&-0.391447368421&. \end{alignedat} \]
2c (130)

Hiroyasu Kamo