Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&, \\ P_{\mathbf{3}}&{}\approx{}&0.000455578038&{}:{}&0.982841274568&{}:{}&0.016703147394&, \\ P^-_{\mathbf{3}}&{}\approx{}&-0.017357695346&{}:{}&1.040822662412&{}:{}&-0.023464967066&, \\ P^+_{\mathbf{3}}&{}\approx{}&0.015570281012&{}:{}&0.933643609634&{}:{}&0.050786109354&, \\ Q_{\mathbf{3}}&{}\approx{}&-0.012581667745&{}:{}&0.784033768415&{}:{}&0.228547899331&, \\ I^\prime_{\mathbf{3}}&{}\approx{}&0.014242493332&{}:{}&0.854025234005&{}:{}&0.131732272663&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{3}}\)
\(P^-_{\mathbf{3}}\)
\(P^+_{\mathbf{3}}\)
\(Q_{\mathbf{3}}\)
\(I^\prime_{\mathbf{3}}\)
3 (022)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3}}&{}\approx{}&-0.321093475754&{}:{}&0.550455614898&{}:{}&0.770637860857&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.016225903608&{}:{}&0.945913654641&{}:{}&0.037860441752&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.739430042465&{}:{}&1.232383404108&{}:{}&-0.971813446573&, \\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.031754982293&{}:{}&0.952064907159&{}:{}&0.016180110547&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006345580218&{}:{}&0.761002375724&{}:{}&0.232652044058&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000429876983&{}:{}&0.927395104842&{}:{}&0.072175018175&, \\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.017246659570&{}:{}&1.005420171254&{}:{}&-0.022666830824&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&1.568548005965&{}:{}&-2.688986610804&{}:{}&2.120438604839&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.016296405516&{}:{}&0.977184346148&{}:{}&0.039112059369&, \\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.774291885177&{}:{}&0.225708114823&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.058257569496&{}:{}&0.000000000000&{}:{}&1.058257569496&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.016309071859&{}:{}&1.016309071859&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3}}}{B^\prime_{\mathbf{3}}}{C^\prime_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3}}}{B^{\prime\prime}_{\mathbf{3}}}{C^{\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3}}}{B^{\prime\prime\prime}_{\mathbf{3}}}{C^{\prime\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^*_{\mathbf{3}}}{B^*_{\mathbf{3}}}{C^*_{\mathbf{3}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3}}}}&{}\approx{}&1.651366844693&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3}}}}&{}\approx{}&0.081129518039&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3}}}}&{}\approx{}&3.697150212324&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.321093475754&{}:{}&0.550455614898&{}:{}&0.770637860857&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.016225903608&{}:{}&0.945913654641&{}:{}&0.037860441752&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.739430042465&{}:{}&1.232383404108&{}:{}&-0.971813446573&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3}}&{}\approx{}&0.000455578038&{}:{}&0.982841274568&{}:{}&0.016703147394&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.031754982293&{}:{}&0.952064907159&{}:{}&0.016180110547&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006345580218&{}:{}&0.761002375724&{}:{}&0.232652044058&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000429876983&{}:{}&0.927395104842&{}:{}&0.072175018175&. \end{alignedat} \]
3 (022)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3}}&{}\approx{}&-0.017357695346&{}:{}&1.040822662412&{}:{}&-0.023464967066&,\\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.017246659570&{}:{}&1.005420171254&{}:{}&-0.022666830824&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&1.568548005965&{}:{}&-2.688986610804&{}:{}&2.120438604839&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.016296405516&{}:{}&0.977184346148&{}:{}&0.039112059369&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3}}&{}\approx{}&0.015570281012&{}:{}&0.933643609634&{}:{}&0.050786109354&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.321093475754&{}:{}&0.550455614898&{}:{}&0.770637860857&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.016225903608&{}:{}&0.945913654641&{}:{}&0.037860441752&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.739430042465&{}:{}&1.232383404108&{}:{}&-0.971813446573&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.031754982293&{}:{}&0.952064907159&{}:{}&0.016180110547&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006345580218&{}:{}&0.761002375724&{}:{}&0.232652044058&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000429876983&{}:{}&0.927395104842&{}:{}&0.072175018175&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3}}&{}\approx{}&-0.012581667745&{}:{}&0.784033768415&{}:{}&0.228547899331&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.774291885177&{}:{}&0.225708114823&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.058257569496&{}:{}&0.000000000000&{}:{}&1.058257569496&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.016309071859&{}:{}&1.016309071859&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3}}&{}\approx{}&0.014242493332&{}:{}&0.854025234005&{}:{}&0.131732272663&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.031754982293&{}:{}&0.952064907159&{}:{}&0.016180110547&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006345580218&{}:{}&0.761002375724&{}:{}&0.232652044058&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000429876983&{}:{}&0.927395104842&{}:{}&0.072175018175&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.774291885177&{}:{}&0.225708114823&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.058257569496&{}:{}&0.000000000000&{}:{}&1.058257569496&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.016309071859&{}:{}&1.016309071859&{}:{}&0.000000000000&. \end{alignedat} \]
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