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,
\[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&, \\ P_{\mathbf{5}}&{}\approx{}&0.996836421159&{}:{}&0.000142612726&{}:{}&0.003020966115&, \\ P^-_{\mathbf{5}}&{}\approx{}&1.035414325931&{}:{}&-0.015988426907&{}:{}&-0.019425899024&, \\ P^+_{\mathbf{5}}&{}\approx{}&0.961664163908&{}:{}&0.014849608325&{}:{}&0.023486227768&, \\ Q_{\mathbf{5}}&{}\approx{}&0.870243223501&{}:{}&-0.013965101424&{}:{}&0.143721877922&, \\ I^\prime_{\mathbf{5}}&{}\approx{}&0.909477399214&{}:{}&0.014043762537&{}:{}&0.076478838249&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{5}}\)
\(P^-_{\mathbf{5}}\)
\(P^+_{\mathbf{5}}\)
\(Q_{\mathbf{5}}\)
\(I^\prime_{\mathbf{5}}\)
5 (202)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5}}&{}\approx{}&0.963942436427&{}:{}&0.015023984822&{}:{}&0.021033578751&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.594492064089&{}:{}&-0.981640213631&{}:{}&1.387148149542&,\\C^\prime_{\mathbf{5}}&{}\approx{}&1.514159796021&{}:{}&2.523599660036&{}:{}&-3.037759456057&, \\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.853773920225&{}:{}&0.006591806573&{}:{}&0.139634273202&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.967198711644&{}:{}&0.029870140906&{}:{}&0.002931147450&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.958424069483&{}:{}&0.000137117250&{}:{}&0.041438813267&, \\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&-0.376299246022&{}:{}&0.621354757393&{}:{}&0.754944488629&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&1.003185612785&{}:{}&0.015635628073&{}:{}&-0.018821240858&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.993661637026&{}:{}&-0.015343699673&{}:{}&0.021682062646&, \\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.107625218511&{}:{}&1.107625218511&,\\B^*_{\mathbf{5}}&{}\approx{}&0.858257569496&{}:{}&0.000000000000&{}:{}&0.141742430504&,\\C^*_{\mathbf{5}}&{}\approx{}&1.016309071859&{}:{}&-0.016309071859&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5}}}{B^\prime_{\mathbf{5}}}{C^\prime_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5}}}{B^{\prime\prime}_{\mathbf{5}}}{C^{\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5}}}{B^{\prime\prime\prime}_{\mathbf{5}}}{C^{\prime\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^*_{\mathbf{5}}}{B^*_{\mathbf{5}}}{C^*_{\mathbf{5}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5}}}}&{}\approx{}&0.045071954466&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5}}}}&{}\approx{}&2.972460320447&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5}}}}&{}\approx{}&7.570798980107&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.963942436427&{}:{}&0.015023984822&{}:{}&0.021033578751&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.594492064089&{}:{}&-0.981640213631&{}:{}&1.387148149542&,\\C^\prime_{\mathbf{5}}&{}\approx{}&1.514159796021&{}:{}&2.523599660036&{}:{}&-3.037759456057&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5}}&{}\approx{}&0.996836421159&{}:{}&0.000142612726&{}:{}&0.003020966115&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.853773920225&{}:{}&0.006591806573&{}:{}&0.139634273202&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.967198711644&{}:{}&0.029870140906&{}:{}&0.002931147450&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.958424069483&{}:{}&0.000137117250&{}:{}&0.041438813267&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5}}&{}\approx{}&1.035414325931&{}:{}&-0.015988426907&{}:{}&-0.019425899024&,\\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&-0.376299246022&{}:{}&0.621354757393&{}:{}&0.754944488629&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&1.003185612785&{}:{}&0.015635628073&{}:{}&-0.018821240858&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.993661637026&{}:{}&-0.015343699673&{}:{}&0.021682062646&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5}}&{}\approx{}&0.961664163908&{}:{}&0.014849608325&{}:{}&0.023486227768&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.963942436427&{}:{}&0.015023984822&{}:{}&0.021033578751&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.594492064089&{}:{}&-0.981640213631&{}:{}&1.387148149542&,\\C^\prime_{\mathbf{5}}&{}\approx{}&1.514159796021&{}:{}&2.523599660036&{}:{}&-3.037759456057&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.853773920225&{}:{}&0.006591806573&{}:{}&0.139634273202&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.967198711644&{}:{}&0.029870140906&{}:{}&0.002931147450&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.958424069483&{}:{}&0.000137117250&{}:{}&0.041438813267&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5}}&{}\approx{}&0.870243223501&{}:{}&-0.013965101424&{}:{}&0.143721877922&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.107625218511&{}:{}&1.107625218511&,\\B^*_{\mathbf{5}}&{}\approx{}&0.858257569496&{}:{}&0.000000000000&{}:{}&0.141742430504&,\\C^*_{\mathbf{5}}&{}\approx{}&1.016309071859&{}:{}&-0.016309071859&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5}}&{}\approx{}&0.909477399214&{}:{}&0.014043762537&{}:{}&0.076478838249&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.853773920225&{}:{}&0.006591806573&{}:{}&0.139634273202&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.967198711644&{}:{}&0.029870140906&{}:{}&0.002931147450&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.958424069483&{}:{}&0.000137117250&{}:{}&0.041438813267&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.107625218511&{}:{}&1.107625218511&,\\B^*_{\mathbf{5}}&{}\approx{}&0.858257569496&{}:{}&0.000000000000&{}:{}&0.141742430504&,\\C^*_{\mathbf{5}}&{}\approx{}&1.016309071859&{}:{}&-0.016309071859&{}:{}&0.000000000000&. \end{alignedat} \]
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