Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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6c(330)

Malfatti circles

6c (330)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&, \\ P_{\mathbf{6c}}&{}\approx{}&0.364883838671&{}:{}&0.884007411079&{}:{}&-0.248891249750&, \\ P^-_{\mathbf{6c}}&{}\approx{}&0.229457479615&{}:{}&0.672474462969&{}:{}&0.098068057416&, \\ P^+_{\mathbf{6c}}&{}\approx{}&0.487687695912&{}:{}&1.075824293278&{}:{}&-0.563511989190&, \\ Q_{\mathbf{6c}}&{}\approx{}&0.379221071192&{}:{}&0.439953570876&{}:{}&0.180825357932&, \\ I^\prime_{\mathbf{6c}}&{}\approx{}&0.572129290421&{}:{}&1.149659403542&{}:{}&-0.721788693962&, \end{alignedat} \]
\(I_{\mathbf{c}}\) Incenter
\(P_{\mathbf{6c}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{6c}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{6c}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{6c}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{6c}}\) Radical center of the Malfatti circles
6c (330)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6c}}&{}\approx{}&1.811435789111&{}:{}&2.028589472778&{}:{}&-2.840025261889&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.330216137896&{}:{}&1.440288183862&{}:{}&-0.770504321758&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.302779483315&{}:{}&0.504632472192&{}:{}&0.192588044493&, \\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.315903201426&{}:{}&0.952182729172&{}:{}&-0.268085930598&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.603312550619&{}:{}&0.808213526449&{}:{}&-0.411526077067&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.646315512321&{}:{}&1.565834499189&{}:{}&-1.212150011510&, \\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.166730025519&{}:{}&0.727218503552&{}:{}&0.106051470930&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.392577105356&{}:{}&0.439638980287&{}:{}&0.167783914357&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.423211653471&{}:{}&1.240312714441&{}:{}&-0.663524367912&, \\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.708712152522&{}:{}&0.291287847478&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.677124344468&{}:{}&0.000000000000&{}:{}&0.322875655532&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.462930676461&{}:{}&0.537069323539&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6c}}}{B^\prime_{\mathbf{6c}}}{C^\prime_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^*_{\mathbf{6c}}}{B^*_{\mathbf{6c}}}{C^*_{\mathbf{6c}}}\)
6c (330)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6c}}}}&{}\approx{}&0.405717894556&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6c}}}}&{}\approx{}&0.110072045965&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6c}}}}&{}\approx{}&0.100926494438&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.811435789111&{}:{}&2.028589472778&{}:{}&-2.840025261889&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.330216137896&{}:{}&1.440288183862&{}:{}&-0.770504321758&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.302779483315&{}:{}&0.504632472192&{}:{}&0.192588044493&. \end{alignedat} \]
6c (330)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6c}}&{}\approx{}&0.364883838671&{}:{}&0.884007411079&{}:{}&-0.248891249750&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.315903201426&{}:{}&0.952182729172&{}:{}&-0.268085930598&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.603312550619&{}:{}&0.808213526449&{}:{}&-0.411526077067&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.646315512321&{}:{}&1.565834499189&{}:{}&-1.212150011510&. \end{alignedat} \]
6c (330)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6c}}&{}\approx{}&0.229457479615&{}:{}&0.672474462969&{}:{}&0.098068057416&,\\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.166730025519&{}:{}&0.727218503552&{}:{}&0.106051470930&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.392577105356&{}:{}&0.439638980287&{}:{}&0.167783914357&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.423211653471&{}:{}&1.240312714441&{}:{}&-0.663524367912&. \end{alignedat} \]
6c (330)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6c}}&{}\approx{}&0.487687695912&{}:{}&1.075824293278&{}:{}&-0.563511989190&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.811435789111&{}:{}&2.028589472778&{}:{}&-2.840025261889&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.330216137896&{}:{}&1.440288183862&{}:{}&-0.770504321758&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.302779483315&{}:{}&0.504632472192&{}:{}&0.192588044493&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.315903201426&{}:{}&0.952182729172&{}:{}&-0.268085930598&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.603312550619&{}:{}&0.808213526449&{}:{}&-0.411526077067&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.646315512321&{}:{}&1.565834499189&{}:{}&-1.212150011510&, \end{alignedat} \]
6c (330)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6c}}&{}\approx{}&0.379221071192&{}:{}&0.439953570876&{}:{}&0.180825357932&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.708712152522&{}:{}&0.291287847478&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.677124344468&{}:{}&0.000000000000&{}:{}&0.322875655532&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.462930676461&{}:{}&0.537069323539&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6c}}&{}\approx{}&0.572129290421&{}:{}&1.149659403542&{}:{}&-0.721788693962&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.315903201426&{}:{}&0.952182729172&{}:{}&-0.268085930598&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.603312550619&{}:{}&0.808213526449&{}:{}&-0.411526077067&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.646315512321&{}:{}&1.565834499189&{}:{}&-1.212150011510&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.708712152522&{}:{}&0.291287847478&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.677124344468&{}:{}&0.000000000000&{}:{}&0.322875655532&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.462930676461&{}:{}&0.537069323539&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)