Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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7a(233)

Malfatti circles

7a (233)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&, \\ P_{\mathbf{7a}}&{}\approx{}&1.000525029026&{}:{}&-0.000307330836&{}:{}&-0.000217698190&, \\ P^-_{\mathbf{7a}}&{}\approx{}&0.986118481558&{}:{}&0.005696353877&{}:{}&0.008185164566&, \\ P^+_{\mathbf{7a}}&{}\approx{}&1.015249647623&{}:{}&-0.006443566305&{}:{}&-0.008806081318&, \\ Q_{\mathbf{7a}}&{}\approx{}&0.914555045413&{}:{}&0.062078979562&{}:{}&0.023365975025&, \\ I^\prime_{\mathbf{7a}}&{}\approx{}&1.047294131480&{}:{}&-0.023696387043&{}:{}&-0.023597744437&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{7a}}\)
\(P^-_{\mathbf{7a}}\)
\(P^+_{\mathbf{7a}}\)
\(Q_{\mathbf{7a}}\)
\(I^\prime_{\mathbf{7a}}\)
7a (233)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7a}}&{}\approx{}&1.014820022545&{}:{}&-0.006175009394&{}:{}&-0.008645013151&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&4.017825533194&{}:{}&6.357100710925&{}:{}&-9.374926244119&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.622334958066&{}:{}&-1.037224930110&{}:{}&1.414889972044&, \\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.077735690145&{}:{}&-0.045503340650&{}:{}&-0.032232349495&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.012497232531&{}:{}&-0.012276929381&{}:{}&-0.000220303150&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.017586668785&{}:{}&-0.000312571652&{}:{}&-0.017274097132&, \\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.205941724766&{}:{}&0.325845976662&{}:{}&0.468212298571&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.997789343342&{}:{}&-0.006071380571&{}:{}&0.008282037230&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.002749764296&{}:{}&0.005792425164&{}:{}&-0.008542189460&, \\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.726537685716&{}:{}&0.273462314284&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.975087481231&{}:{}&0.000000000000&{}:{}&0.024912518769&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.936435780472&{}:{}&0.063564219528&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7a}}}{B^\prime_{\mathbf{7a}}}{C^\prime_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^*_{\mathbf{7a}}}{B^*_{\mathbf{7a}}}{C^*_{\mathbf{7a}}}\)
7a (233)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7a}}}}&{}\approx{}&-0.011115016909&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7a}}}}&{}\approx{}&-12.053476599582&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7a}}}}&{}\approx{}&-1.867004874198&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.014820022545&{}:{}&-0.006175009394&{}:{}&-0.008645013151&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&4.017825533194&{}:{}&6.357100710925&{}:{}&-9.374926244119&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.622334958066&{}:{}&-1.037224930110&{}:{}&1.414889972044&. \end{alignedat} \]
7a (233)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7a}}&{}\approx{}&1.000525029026&{}:{}&-0.000307330836&{}:{}&-0.000217698190&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.077735690145&{}:{}&-0.045503340650&{}:{}&-0.032232349495&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.012497232531&{}:{}&-0.012276929381&{}:{}&-0.000220303150&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.017586668785&{}:{}&-0.000312571652&{}:{}&-0.017274097132&. \end{alignedat} \]
7a (233)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7a}}&{}\approx{}&0.986118481558&{}:{}&0.005696353877&{}:{}&0.008185164566&,\\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.205941724766&{}:{}&0.325845976662&{}:{}&0.468212298571&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.997789343342&{}:{}&-0.006071380571&{}:{}&0.008282037230&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.002749764296&{}:{}&0.005792425164&{}:{}&-0.008542189460&. \end{alignedat} \]
7a (233)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7a}}&{}\approx{}&1.015249647623&{}:{}&-0.006443566305&{}:{}&-0.008806081318&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.014820022545&{}:{}&-0.006175009394&{}:{}&-0.008645013151&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&4.017825533194&{}:{}&6.357100710925&{}:{}&-9.374926244119&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.622334958066&{}:{}&-1.037224930110&{}:{}&1.414889972044&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.077735690145&{}:{}&-0.045503340650&{}:{}&-0.032232349495&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.012497232531&{}:{}&-0.012276929381&{}:{}&-0.000220303150&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.017586668785&{}:{}&-0.000312571652&{}:{}&-0.017274097132&, \end{alignedat} \]
7a (233)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7a}}&{}\approx{}&0.914555045413&{}:{}&0.062078979562&{}:{}&0.023365975025&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.726537685716&{}:{}&0.273462314284&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.975087481231&{}:{}&0.000000000000&{}:{}&0.024912518769&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.936435780472&{}:{}&0.063564219528&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7a}}&{}\approx{}&1.047294131480&{}:{}&-0.023696387043&{}:{}&-0.023597744437&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.077735690145&{}:{}&-0.045503340650&{}:{}&-0.032232349495&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.012497232531&{}:{}&-0.012276929381&{}:{}&-0.000220303150&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.017586668785&{}:{}&-0.000312571652&{}:{}&-0.017274097132&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.726537685716&{}:{}&0.273462314284&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.975087481231&{}:{}&0.000000000000&{}:{}&0.024912518769&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.936435780472&{}:{}&0.063564219528&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)