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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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&, \\ P_{\mathbf{1}}&{}\approx{}&0.008641558567&{}:{}&0.009587605817&{}:{}&0.981770835617&, \\ P^-_{\mathbf{1}}&{}\approx{}&-0.014273081925&{}:{}&-0.015193285258&{}:{}&1.029466367183&, \\ P^+_{\mathbf{1}}&{}\approx{}&0.028441727718&{}:{}&0.031000371825&{}:{}&0.940557900457&, \\ Q_{\mathbf{1}}&{}\approx{}&-0.087912087912&{}:{}&-0.098901098901&{}:{}&1.186813186813&, \\ I^\prime_{\mathbf{1}}&{}\approx{}&0.071337579618&{}:{}&0.078170237406&{}:{}&0.850492182976&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{1}}\)
\(P^-_{\mathbf{1}}\)
\(P^+_{\mathbf{1}}\)
\(Q_{\mathbf{1}}\)
\(I^\prime_{\mathbf{1}}\)
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1}}&{}\approx{}&-0.350000000000&{}:{}&0.626159554731&{}:{}&0.723840445269&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.622222222222&{}:{}&-0.400673400673&{}:{}&0.778451178451&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.021875000000&{}:{}&0.023674242424&{}:{}&0.954450757576&, \\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.042264150943&{}:{}&0.009262435678&{}:{}&0.948473413379&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.008327137546&{}:{}&0.045623521460&{}:{}&0.946049340994&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.118440185063&{}:{}&0.131406597368&{}:{}&0.750153217569&, \\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.022733423545&{}:{}&-0.014638946980&{}:{}&0.991905523435&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.013714514194&{}:{}&0.024535640289&{}:{}&0.989178873905&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-274.400000000094&{}:{}&-292.090909091008&{}:{}&567.490909091102&, \\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.090909090909&{}:{}&1.090909090909&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.080000000000&{}:{}&0.000000000000&{}:{}&1.080000000000&,\\C^*_{\mathbf{1}}&{}\approx{}&0.470588235294&{}:{}&0.529411764706&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1}}}{B^\prime_{\mathbf{1}}}{C^\prime_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1}}}{B^{\prime\prime}_{\mathbf{1}}}{C^{\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1}}}{B^{\prime\prime\prime}_{\mathbf{1}}}{C^{\prime\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^*_{\mathbf{1}}}{B^*_{\mathbf{1}}}{C^*_{\mathbf{1}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1}}}}&{}\approx{}&1.928571428571&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1}}}}&{}\approx{}&2.074074074074&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1}}}}&{}\approx{}&0.072916666667&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.350000000000&{}:{}&0.626159554731&{}:{}&0.723840445269&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.622222222222&{}:{}&-0.400673400673&{}:{}&0.778451178451&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.021875000000&{}:{}&0.023674242424&{}:{}&0.954450757576&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1}}&{}\approx{}&0.008641558567&{}:{}&0.009587605817&{}:{}&0.981770835617&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.042264150943&{}:{}&0.009262435678&{}:{}&0.948473413379&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.008327137546&{}:{}&0.045623521460&{}:{}&0.946049340994&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.118440185063&{}:{}&0.131406597368&{}:{}&0.750153217569&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1}}&{}\approx{}&-0.014273081925&{}:{}&-0.015193285258&{}:{}&1.029466367183&,\\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.022733423545&{}:{}&-0.014638946980&{}:{}&0.991905523435&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.013714514194&{}:{}&0.024535640289&{}:{}&0.989178873905&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-274.400000000094&{}:{}&-292.090909091008&{}:{}&567.490909091102&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1}}&{}\approx{}&0.028441727718&{}:{}&0.031000371825&{}:{}&0.940557900457&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.350000000000&{}:{}&0.626159554731&{}:{}&0.723840445269&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.622222222222&{}:{}&-0.400673400673&{}:{}&0.778451178451&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.021875000000&{}:{}&0.023674242424&{}:{}&0.954450757576&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.042264150943&{}:{}&0.009262435678&{}:{}&0.948473413379&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.008327137546&{}:{}&0.045623521460&{}:{}&0.946049340994&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.118440185063&{}:{}&0.131406597368&{}:{}&0.750153217569&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1}}&{}\approx{}&-0.087912087912&{}:{}&-0.098901098901&{}:{}&1.186813186813&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.090909090909&{}:{}&1.090909090909&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.080000000000&{}:{}&0.000000000000&{}:{}&1.080000000000&,\\C^*_{\mathbf{1}}&{}\approx{}&0.470588235294&{}:{}&0.529411764706&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1}}&{}\approx{}&0.071337579618&{}:{}&0.078170237406&{}:{}&0.850492182976&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.042264150943&{}:{}&0.009262435678&{}:{}&0.948473413379&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.008327137546&{}:{}&0.045623521460&{}:{}&0.946049340994&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.118440185063&{}:{}&0.131406597368&{}:{}&0.750153217569&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.090909090909&{}:{}&1.090909090909&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.080000000000&{}:{}&0.000000000000&{}:{}&1.080000000000&,\\C^*_{\mathbf{1}}&{}\approx{}&0.470588235294&{}:{}&0.529411764706&{}:{}&0.000000000000&. \end{alignedat} \]
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