Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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6a(231)

Malfatti circles

6a (231)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&, \\ P_{\mathbf{6a}}&{}\approx{}&1.001415569936&{}:{}&-0.001197677978&{}:{}&-0.000217891958&, \\ P^-_{\mathbf{6a}}&{}\approx{}&0.973240786545&{}:{}&0.010554646162&{}:{}&0.016204567293&, \\ P^+_{\mathbf{6a}}&{}\approx{}&1.030832247618&{}:{}&-0.013468023643&{}:{}&-0.017364223975&, \\ Q_{\mathbf{6a}}&{}\approx{}&1.121567151553&{}:{}&-0.150222080727&{}:{}&0.028654929174&, \\ I^\prime_{\mathbf{6a}}&{}\approx{}&1.072016560229&{}:{}&-0.047861767358&{}:{}&-0.024154792871&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{6a}}\)
\(P^-_{\mathbf{6a}}\)
\(P^+_{\mathbf{6a}}\)
\(Q_{\mathbf{6a}}\)
\(I^\prime_{\mathbf{6a}}\)
6a (231)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6a}}&{}\approx{}&1.029779944088&{}:{}&-0.012408310037&{}:{}&-0.017371634051&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&5.998426136278&{}:{}&8.997901515037&{}:{}&-13.996327651315&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.611115016909&{}:{}&-1.018525028182&{}:{}&1.407410011273&, \\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.109223228887&{}:{}&-0.092411015959&{}:{}&-0.016812212927&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.024740894495&{}:{}&-0.024517927321&{}:{}&-0.000222967174&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.035939167404&{}:{}&-0.001238967682&{}:{}&-0.034700199722&, \\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.208200450502&{}:{}&0.312309780339&{}:{}&0.479489769159&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.995420435007&{}:{}&-0.011994295913&{}:{}&0.016573860905&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.006060936125&{}:{}&0.010910575620&{}:{}&-0.016971511745&, \\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.235712762933&{}:{}&-0.235712762933&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.975087481231&{}:{}&0.000000000000&{}:{}&0.024912518769&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.154653670708&{}:{}&-0.154653670708&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6a}}}{B^\prime_{\mathbf{6a}}}{C^\prime_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^*_{\mathbf{6a}}}{B^*_{\mathbf{6a}}}{C^*_{\mathbf{6a}}}\)
6a (231)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6a}}}}&{}\approx{}&-0.022334958066&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6a}}}}&{}\approx{}&-17.995278408834&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6a}}}}&{}\approx{}&-1.833345050727&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.029779944088&{}:{}&-0.012408310037&{}:{}&-0.017371634051&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&5.998426136278&{}:{}&8.997901515037&{}:{}&-13.996327651315&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.611115016909&{}:{}&-1.018525028182&{}:{}&1.407410011273&. \end{alignedat} \]
6a (231)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6a}}&{}\approx{}&1.001415569936&{}:{}&-0.001197677978&{}:{}&-0.000217891958&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.109223228887&{}:{}&-0.092411015959&{}:{}&-0.016812212927&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.024740894495&{}:{}&-0.024517927321&{}:{}&-0.000222967174&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.035939167404&{}:{}&-0.001238967682&{}:{}&-0.034700199722&. \end{alignedat} \]
6a (231)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6a}}&{}\approx{}&0.973240786545&{}:{}&0.010554646162&{}:{}&0.016204567293&,\\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.208200450502&{}:{}&0.312309780339&{}:{}&0.479489769159&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.995420435007&{}:{}&-0.011994295913&{}:{}&0.016573860905&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.006060936125&{}:{}&0.010910575620&{}:{}&-0.016971511745&. \end{alignedat} \]
6a (231)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6a}}&{}\approx{}&1.030832247618&{}:{}&-0.013468023643&{}:{}&-0.017364223975&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.029779944088&{}:{}&-0.012408310037&{}:{}&-0.017371634051&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&5.998426136278&{}:{}&8.997901515037&{}:{}&-13.996327651315&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.611115016909&{}:{}&-1.018525028182&{}:{}&1.407410011273&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.109223228887&{}:{}&-0.092411015959&{}:{}&-0.016812212927&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.024740894495&{}:{}&-0.024517927321&{}:{}&-0.000222967174&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.035939167404&{}:{}&-0.001238967682&{}:{}&-0.034700199722&, \end{alignedat} \]
6a (231)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6a}}&{}\approx{}&1.121567151553&{}:{}&-0.150222080727&{}:{}&0.028654929174&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.235712762933&{}:{}&-0.235712762933&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.975087481231&{}:{}&0.000000000000&{}:{}&0.024912518769&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.154653670708&{}:{}&-0.154653670708&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6a}}&{}\approx{}&1.072016560229&{}:{}&-0.047861767358&{}:{}&-0.024154792871&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.109223228887&{}:{}&-0.092411015959&{}:{}&-0.016812212927&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.024740894495&{}:{}&-0.024517927321&{}:{}&-0.000222967174&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.035939167404&{}:{}&-0.001238967682&{}:{}&-0.034700199722&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.235712762933&{}:{}&-0.235712762933&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.975087481231&{}:{}&0.000000000000&{}:{}&0.024912518769&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.154653670708&{}:{}&-0.154653670708&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)