Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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3c(132)

Malfatti circles

3c (132)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&, \\ P_{\mathbf{3c}}&{}\approx{}&-0.143116987592&{}:{}&-0.745216502540&{}:{}&1.888333490132&, \\ P^-_{\mathbf{3c}}&{}\approx{}&0.101306788342&{}:{}&-0.298441051084&{}:{}&1.197134262742&, \\ P^+_{\mathbf{3c}}&{}\approx{}&-0.432557280265&{}:{}&-1.274276398347&{}:{}&2.706833678612&, \\ Q_{\mathbf{3c}}&{}\approx{}&0.716156161129&{}:{}&-1.218055616328&{}:{}&1.501899455199&, \\ I^\prime_{\mathbf{3c}}&{}\approx{}&-0.623958723754&{}:{}&-1.838128524240&{}:{}&3.462087247994&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{3c}}\)
\(P^-_{\mathbf{3c}}\)
\(P^+_{\mathbf{3c}}\)
\(Q_{\mathbf{3c}}\)
\(I^\prime_{\mathbf{3c}}\)
3c (132)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3c}}&{}\approx{}&0.394441033370&{}:{}&-1.513897416576&{}:{}&2.119456383207&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-18.488954898986&{}:{}&-23.651939865314&{}:{}&43.140894764300&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.405717894556&{}:{}&-0.676196490926&{}:{}&2.081914385482&, \\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.794012146822&{}:{}&-1.169545612637&{}:{}&2.963557759459&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.158666999892&{}:{}&-0.934838563044&{}:{}&2.093505562936&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.489832575446&{}:{}&-2.550579947535&{}:{}&4.040412522980&, \\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.350608210434&{}:{}&-0.448514497157&{}:{}&1.799122707591&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.111372992221&{}:{}&-0.427458785813&{}:{}&1.316085793593&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.291766139272&{}:{}&-0.859517853643&{}:{}&1.567751714371&, \\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-4.291287847478&{}:{}&5.291287847478&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.322875655532&{}:{}&0.000000000000&{}:{}&0.677124344468&,\\C^*_{\mathbf{3c}}&{}\approx{}&-1.426891688585&{}:{}&2.426891688585&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3c}}}{B^\prime_{\mathbf{3c}}}{C^\prime_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^*_{\mathbf{3c}}}{B^*_{\mathbf{3c}}}{C^*_{\mathbf{3c}}}\)
3c (132)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.302779483315&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3c}}}}&{}\approx{}&-6.162984966329&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.135239298185&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.394441033370&{}:{}&-1.513897416576&{}:{}&2.119456383207&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-18.488954898986&{}:{}&-23.651939865314&{}:{}&43.140894764300&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.405717894556&{}:{}&-0.676196490926&{}:{}&2.081914385482&. \end{alignedat} \]
3c (132)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3c}}&{}\approx{}&-0.143116987592&{}:{}&-0.745216502540&{}:{}&1.888333490132&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.794012146822&{}:{}&-1.169545612637&{}:{}&2.963557759459&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.158666999892&{}:{}&-0.934838563044&{}:{}&2.093505562936&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.489832575446&{}:{}&-2.550579947535&{}:{}&4.040412522980&. \end{alignedat} \]
3c (132)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3c}}&{}\approx{}&0.101306788342&{}:{}&-0.298441051084&{}:{}&1.197134262742&,\\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.350608210434&{}:{}&-0.448514497157&{}:{}&1.799122707591&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.111372992221&{}:{}&-0.427458785813&{}:{}&1.316085793593&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.291766139272&{}:{}&-0.859517853643&{}:{}&1.567751714371&. \end{alignedat} \]
3c (132)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3c}}&{}\approx{}&-0.432557280265&{}:{}&-1.274276398347&{}:{}&2.706833678612&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.394441033370&{}:{}&-1.513897416576&{}:{}&2.119456383207&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-18.488954898986&{}:{}&-23.651939865314&{}:{}&43.140894764300&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.405717894556&{}:{}&-0.676196490926&{}:{}&2.081914385482&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.794012146822&{}:{}&-1.169545612637&{}:{}&2.963557759459&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.158666999892&{}:{}&-0.934838563044&{}:{}&2.093505562936&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.489832575446&{}:{}&-2.550579947535&{}:{}&4.040412522980&, \end{alignedat} \]
3c (132)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3c}}&{}\approx{}&0.716156161129&{}:{}&-1.218055616328&{}:{}&1.501899455199&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-4.291287847478&{}:{}&5.291287847478&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.322875655532&{}:{}&0.000000000000&{}:{}&0.677124344468&,\\C^*_{\mathbf{3c}}&{}\approx{}&-1.426891688585&{}:{}&2.426891688585&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3c}}&{}\approx{}&-0.623958723754&{}:{}&-1.838128524240&{}:{}&3.462087247994&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.794012146822&{}:{}&-1.169545612637&{}:{}&2.963557759459&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.158666999892&{}:{}&-0.934838563044&{}:{}&2.093505562936&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.489832575446&{}:{}&-2.550579947535&{}:{}&4.040412522980&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-4.291287847478&{}:{}&5.291287847478&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.322875655532&{}:{}&0.000000000000&{}:{}&0.677124344468&,\\C^*_{\mathbf{3c}}&{}\approx{}&-1.426891688585&{}:{}&2.426891688585&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)