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”.

\(a:b:c=231:250:289\).

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1a (013)

Malfatti circles

1a (013)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&, \\ P_{\mathbf{1a}}&{}\approx{}&-0.067562380038&{}:{}&0.947628187551&{}:{}&0.119934192487&, \\ P^-_{\mathbf{1a}}&{}\approx{}&0.009397693293&{}:{}&0.962958442668&{}:{}&0.027643864039&, \\ P^+_{\mathbf{1a}}&{}\approx{}&-0.130359010108&{}:{}&0.935119255091&{}:{}&0.195239755017&, \\ Q_{\mathbf{1a}}&{}\approx{}&0.160000000000&{}:{}&1.200000000000&{}:{}&-0.360000000000&, \\ I^\prime_{\mathbf{1a}}&{}\approx{}&-0.227979274611&{}:{}&0.888230940044&{}:{}&0.339748334567&, \end{alignedat} \]
\(I_{\mathbf{a}}\) Incenter
\(P_{\mathbf{1a}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{1a}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{1a}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{1a}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{1a}}\) Radical center of the Malfatti circles
1a (013)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1a}}&{}\approx{}&0.045454545455&{}:{}&0.442739079103&{}:{}&0.511806375443&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.085937500000&{}:{}&0.978422619048&{}:{}&0.107514880952&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-2.444444444444&{}:{}&2.645502645503&{}:{}&0.798941798942&, \\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.166037735849&{}:{}&1.035040431267&{}:{}&0.130997304582&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.311504424779&{}:{}&0.758533501896&{}:{}&0.552970922882&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.063127690100&{}:{}&0.885427341668&{}:{}&0.177700348432&, \\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.093352192362&{}:{}&1.062840977975&{}:{}&0.030511214387&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&0.073089700997&{}:{}&0.711912672046&{}:{}&0.214997626958&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&0.008716323296&{}:{}&0.893140140367&{}:{}&0.098143536337&, \\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.428571428571&{}:{}&-0.428571428571&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.800000000000&{}:{}&0.000000000000&{}:{}&1.800000000000&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.117647058824&{}:{}&0.882352941176&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1a}}}{B^\prime_{\mathbf{1a}}}{C^\prime_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^*_{\mathbf{1a}}}{B^*_{\mathbf{1a}}}{C^*_{\mathbf{1a}}}\)
1a (013)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1a}}}}&{}\approx{}&0.545454545455&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1a}}}}&{}\approx{}&0.114583333333&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1a}}}}&{}\approx{}&3.259259259259&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&0.045454545455&{}:{}&0.442739079103&{}:{}&0.511806375443&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.085937500000&{}:{}&0.978422619048&{}:{}&0.107514880952&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-2.444444444444&{}:{}&2.645502645503&{}:{}&0.798941798942&. \end{alignedat} \]
1a (013)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1a}}&{}\approx{}&-0.067562380038&{}:{}&0.947628187551&{}:{}&0.119934192487&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.166037735849&{}:{}&1.035040431267&{}:{}&0.130997304582&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.311504424779&{}:{}&0.758533501896&{}:{}&0.552970922882&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.063127690100&{}:{}&0.885427341668&{}:{}&0.177700348432&. \end{alignedat} \]
1a (013)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1a}}&{}\approx{}&0.009397693293&{}:{}&0.962958442668&{}:{}&0.027643864039&,\\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.093352192362&{}:{}&1.062840977975&{}:{}&0.030511214387&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&0.073089700997&{}:{}&0.711912672046&{}:{}&0.214997626958&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&0.008716323296&{}:{}&0.893140140367&{}:{}&0.098143536337&. \end{alignedat} \]
1a (013)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1a}}&{}\approx{}&-0.130359010108&{}:{}&0.935119255091&{}:{}&0.195239755017&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&0.045454545455&{}:{}&0.442739079103&{}:{}&0.511806375443&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.085937500000&{}:{}&0.978422619048&{}:{}&0.107514880952&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-2.444444444444&{}:{}&2.645502645503&{}:{}&0.798941798942&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.166037735849&{}:{}&1.035040431267&{}:{}&0.130997304582&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.311504424779&{}:{}&0.758533501896&{}:{}&0.552970922882&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.063127690100&{}:{}&0.885427341668&{}:{}&0.177700348432&, \end{alignedat} \]
1a (013)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1a}}&{}\approx{}&0.160000000000&{}:{}&1.200000000000&{}:{}&-0.360000000000&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.428571428571&{}:{}&-0.428571428571&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.800000000000&{}:{}&0.000000000000&{}:{}&1.800000000000&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.117647058824&{}:{}&0.882352941176&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1a}}&{}\approx{}&-0.227979274611&{}:{}&0.888230940044&{}:{}&0.339748334567&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.166037735849&{}:{}&1.035040431267&{}:{}&0.130997304582&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.311504424779&{}:{}&0.758533501896&{}:{}&0.552970922882&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.063127690100&{}:{}&0.885427341668&{}:{}&0.177700348432&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.428571428571&{}:{}&-0.428571428571&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.800000000000&{}:{}&0.000000000000&{}:{}&1.800000000000&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.117647058824&{}:{}&0.882352941176&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)