Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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6(220)

Malfatti circles

6 (220)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&, \\ P_{\mathbf{6}}&{}\approx{}&0.848421382210&{}:{}&0.151292930412&{}:{}&0.000285687378&, \\ P^-_{\mathbf{6}}&{}\approx{}&1.406431713679&{}:{}&-0.005365113819&{}:{}&-0.401066599860&, \\ P^+_{\mathbf{6}}&{}\approx{}&0.643356086122&{}:{}&0.208863778859&{}:{}&0.147780135019&, \\ Q_{\mathbf{6}}&{}\approx{}&0.661593775481&{}:{}&0.374827268378&{}:{}&-0.036421043859&, \\ I^\prime_{\mathbf{6}}&{}\approx{}&0.635649016855&{}:{}&0.346533535314&{}:{}&0.017817447831&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{6}}\)
\(P^-_{\mathbf{6}}\)
\(P^+_{\mathbf{6}}\)
\(Q_{\mathbf{6}}\)
\(I^\prime_{\mathbf{6}}\)
6 (220)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6}}&{}\approx{}&0.671364425571&{}:{}&0.136931489345&{}:{}&0.191704085083&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.302831959204&{}:{}&-0.009439864014&{}:{}&0.706607904810&,\\C^\prime_{\mathbf{6}}&{}\approx{}&2.972460320447&{}:{}&4.954100534079&{}:{}&-6.926560854526&, \\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.549664521906&{}:{}&0.449486709557&{}:{}&0.000848768538&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.733256098333&{}:{}&0.266496993685&{}:{}&0.000246907983&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.592717830532&{}:{}&0.105695140845&{}:{}&0.301587028623&, \\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&-0.734528818062&{}:{}&0.022896698800&{}:{}&1.711632119261&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.088383327267&{}:{}&0.221986665222&{}:{}&-0.310369992490&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.780195941150&{}:{}&-0.002976212769&{}:{}&0.222780271619&, \\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.107625218511&{}:{}&-0.107625218511&,\\B^*_{\mathbf{6}}&{}\approx{}&1.058257569496&{}:{}&0.000000000000&{}:{}&-0.058257569496&,\\C^*_{\mathbf{6}}&{}\approx{}&0.638344598849&{}:{}&0.361655401151&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6}}}{B^\prime_{\mathbf{6}}}{C^\prime_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6}}}{B^{\prime\prime}_{\mathbf{6}}}{C^{\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6}}}{B^{\prime\prime\prime}_{\mathbf{6}}}{C^{\prime\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^*_{\mathbf{6}}}{B^*_{\mathbf{6}}}{C^*_{\mathbf{6}}}\)
6 (220)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6}}}}&{}\approx{}&0.410794468036&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6}}}}&{}\approx{}&1.514159796021&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6}}}}&{}\approx{}&14.862301602236&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.671364425571&{}:{}&0.136931489345&{}:{}&0.191704085083&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.302831959204&{}:{}&-0.009439864014&{}:{}&0.706607904810&,\\C^\prime_{\mathbf{6}}&{}\approx{}&2.972460320447&{}:{}&4.954100534079&{}:{}&-6.926560854526&. \end{alignedat} \]
6 (220)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6}}&{}\approx{}&0.848421382210&{}:{}&0.151292930412&{}:{}&0.000285687378&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.549664521906&{}:{}&0.449486709557&{}:{}&0.000848768538&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.733256098333&{}:{}&0.266496993685&{}:{}&0.000246907983&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.592717830532&{}:{}&0.105695140845&{}:{}&0.301587028623&. \end{alignedat} \]
6 (220)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6}}&{}\approx{}&1.406431713679&{}:{}&-0.005365113819&{}:{}&-0.401066599860&,\\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&-0.734528818062&{}:{}&0.022896698800&{}:{}&1.711632119261&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.088383327267&{}:{}&0.221986665222&{}:{}&-0.310369992490&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.780195941150&{}:{}&-0.002976212769&{}:{}&0.222780271619&. \end{alignedat} \]
6 (220)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6}}&{}\approx{}&0.643356086122&{}:{}&0.208863778859&{}:{}&0.147780135019&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.671364425571&{}:{}&0.136931489345&{}:{}&0.191704085083&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.302831959204&{}:{}&-0.009439864014&{}:{}&0.706607904810&,\\C^\prime_{\mathbf{6}}&{}\approx{}&2.972460320447&{}:{}&4.954100534079&{}:{}&-6.926560854526&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.549664521906&{}:{}&0.449486709557&{}:{}&0.000848768538&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.733256098333&{}:{}&0.266496993685&{}:{}&0.000246907983&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.592717830532&{}:{}&0.105695140845&{}:{}&0.301587028623&, \end{alignedat} \]
6 (220)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6}}&{}\approx{}&0.661593775481&{}:{}&0.374827268378&{}:{}&-0.036421043859&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.107625218511&{}:{}&-0.107625218511&,\\B^*_{\mathbf{6}}&{}\approx{}&1.058257569496&{}:{}&0.000000000000&{}:{}&-0.058257569496&,\\C^*_{\mathbf{6}}&{}\approx{}&0.638344598849&{}:{}&0.361655401151&{}:{}&0.000000000000&. \end{alignedat} \]
6 (220)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6}}&{}\approx{}&0.635649016855&{}:{}&0.346533535314&{}:{}&0.017817447831&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.549664521906&{}:{}&0.449486709557&{}:{}&0.000848768538&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.733256098333&{}:{}&0.266496993685&{}:{}&0.000246907983&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.592717830532&{}:{}&0.105695140845&{}:{}&0.301587028623&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.107625218511&{}:{}&-0.107625218511&,\\B^*_{\mathbf{6}}&{}\approx{}&1.058257569496&{}:{}&0.000000000000&{}:{}&-0.058257569496&,\\C^*_{\mathbf{6}}&{}\approx{}&0.638344598849&{}:{}&0.361655401151&{}:{}&0.000000000000&. \end{alignedat} \]
6 (220)