Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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4a(211)

Malfatti circles

4a (211)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&, \\ P_{\mathbf{4a}}&{}\approx{}&1.045375736184&{}:{}&-0.001250253677&{}:{}&-0.044125482507&, \\ P^-_{\mathbf{4a}}&{}\approx{}&0.716287621340&{}:{}&0.131655359368&{}:{}&0.152057019292&, \\ P^+_{\mathbf{4a}}&{}\approx{}&1.675071904802&{}:{}&-0.255559514775&{}:{}&-0.419512390027&, \\ Q_{\mathbf{4a}}&{}\approx{}&1.959982000059&{}:{}&-0.262518899407&{}:{}&-0.697463100652&, \\ I^\prime_{\mathbf{4a}}&{}\approx{}&1.558793627719&{}:{}&-0.069594650621&{}:{}&-0.489198977098&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{4a}}\)
\(P^-_{\mathbf{4a}}\)
\(P^+_{\mathbf{4a}}\)
\(Q_{\mathbf{4a}}\)
\(I^\prime_{\mathbf{4a}}\)
4a (211)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4a}}&{}\approx{}&1.799580303007&{}:{}&-0.333158459586&{}:{}&-0.466421843421&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&3.111674790330&{}:{}&5.148899720440&{}:{}&-7.260574510770&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.803565106639&{}:{}&-1.339275177731&{}:{}&1.535710071093&, \\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.277282286918&{}:{}&-0.007640056732&{}:{}&-0.269642230186&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.601168311558&{}:{}&-0.533582732635&{}:{}&-0.067585578923&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.878781780718&{}:{}&-0.002246994786&{}:{}&-0.876534785932&, \\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.219018488896&{}:{}&0.362410699137&{}:{}&0.418570811967&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.973564166329&{}:{}&-0.180237101629&{}:{}&0.206672935300&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.081526919385&{}:{}&0.198787206417&{}:{}&-0.280314125803&, \\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.273462314284&{}:{}&0.726537685716&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.552437750421&{}:{}&0.000000000000&{}:{}&-0.552437750421&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.154653670708&{}:{}&-0.154653670708&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4a}}}{B^\prime_{\mathbf{4a}}}{C^\prime_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^*_{\mathbf{4a}}}{B^*_{\mathbf{4a}}}{C^*_{\mathbf{4a}}}\)
4a (211)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4a}}}}&{}\approx{}&-0.599685227256&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4a}}}}&{}\approx{}&-9.335024370989&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4a}}}}&{}\approx{}&-2.410695319916&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&1.799580303007&{}:{}&-0.333158459586&{}:{}&-0.466421843421&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&3.111674790330&{}:{}&5.148899720440&{}:{}&-7.260574510770&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.803565106639&{}:{}&-1.339275177731&{}:{}&1.535710071093&. \end{alignedat} \]
4a (211)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4a}}&{}\approx{}&1.045375736184&{}:{}&-0.001250253677&{}:{}&-0.044125482507&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.277282286918&{}:{}&-0.007640056732&{}:{}&-0.269642230186&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.601168311558&{}:{}&-0.533582732635&{}:{}&-0.067585578923&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.878781780718&{}:{}&-0.002246994786&{}:{}&-0.876534785932&. \end{alignedat} \]
4a (211)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4a}}&{}\approx{}&0.716287621340&{}:{}&0.131655359368&{}:{}&0.152057019292&,\\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.219018488896&{}:{}&0.362410699137&{}:{}&0.418570811967&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.973564166329&{}:{}&-0.180237101629&{}:{}&0.206672935300&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.081526919385&{}:{}&0.198787206417&{}:{}&-0.280314125803&. \end{alignedat} \]
4a (211)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4a}}&{}\approx{}&1.675071904802&{}:{}&-0.255559514775&{}:{}&-0.419512390027&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&1.799580303007&{}:{}&-0.333158459586&{}:{}&-0.466421843421&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&3.111674790330&{}:{}&5.148899720440&{}:{}&-7.260574510770&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.803565106639&{}:{}&-1.339275177731&{}:{}&1.535710071093&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.277282286918&{}:{}&-0.007640056732&{}:{}&-0.269642230186&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.601168311558&{}:{}&-0.533582732635&{}:{}&-0.067585578923&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.878781780718&{}:{}&-0.002246994786&{}:{}&-0.876534785932&, \end{alignedat} \]
4a (211)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4a}}&{}\approx{}&1.959982000059&{}:{}&-0.262518899407&{}:{}&-0.697463100652&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.273462314284&{}:{}&0.726537685716&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.552437750421&{}:{}&0.000000000000&{}:{}&-0.552437750421&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.154653670708&{}:{}&-0.154653670708&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4a}}&{}\approx{}&1.558793627719&{}:{}&-0.069594650621&{}:{}&-0.489198977098&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.277282286918&{}:{}&-0.007640056732&{}:{}&-0.269642230186&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.601168311558&{}:{}&-0.533582732635&{}:{}&-0.067585578923&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.878781780718&{}:{}&-0.002246994786&{}:{}&-0.876534785932&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.273462314284&{}:{}&0.726537685716&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.552437750421&{}:{}&0.000000000000&{}:{}&-0.552437750421&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.154653670708&{}:{}&-0.154653670708&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)