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,
\[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&, \\ P_{\mathbf{1}}&{}\approx{}&0.025384292556&{}:{}&0.043935242169&{}:{}&0.930680465275&, \\ P^-_{\mathbf{1}}&{}\approx{}&-0.000555688782&{}:{}&0.000943800024&{}:{}&0.999611888757&, \\ P^+_{\mathbf{1}}&{}\approx{}&0.045382591818&{}:{}&0.077079280111&{}:{}&0.877538128072&, \\ Q_{\mathbf{1}}&{}\approx{}&-0.064934692603&{}:{}&-0.114613828991&{}:{}&1.179548521594&, \\ I^\prime_{\mathbf{1}}&{}\approx{}&0.083698291999&{}:{}&0.142155920044&{}:{}&0.774145787956&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{1}}\)
\(P^-_{\mathbf{1}}\)
\(P^+_{\mathbf{1}}\)
\(Q_{\mathbf{1}}\)
\(I^\prime_{\mathbf{1}}\)
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1}}&{}\approx{}&-0.004697758905&{}:{}&0.418624066210&{}:{}&0.586073692694&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.297168040796&{}:{}&0.009439864014&{}:{}&0.693392095190&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.027539679553&{}:{}&0.045899465921&{}:{}&0.926560854526&, \\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.050407872089&{}:{}&0.042807190345&{}:{}&0.906784937566&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.024354305496&{}:{}&0.082728299997&{}:{}&0.892917394507&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.133575265356&{}:{}&0.231192640811&{}:{}&0.635232093834&, \\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.028838104788&{}:{}&0.000916073569&{}:{}&0.970245821643&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.000529946914&{}:{}&0.047224333205&{}:{}&0.953305613708&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.007971020534&{}:{}&0.013538242306&{}:{}&0.994432778228&, \\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.107625218511&{}:{}&1.107625218511&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.058257569496&{}:{}&0.000000000000&{}:{}&1.058257569496&,\\C^*_{\mathbf{1}}&{}\approx{}&0.361655401151&{}:{}&0.638344598849&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1}}}{B^\prime_{\mathbf{1}}}{C^\prime_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1}}}{B^{\prime\prime}_{\mathbf{1}}}{C^{\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1}}}{B^{\prime\prime\prime}_{\mathbf{1}}}{C^{\prime\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^*_{\mathbf{1}}}{B^*_{\mathbf{1}}}{C^*_{\mathbf{1}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1}}}}&{}\approx{}&1.255872198631&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1}}}}&{}\approx{}&1.485840203979&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1}}}}&{}\approx{}&0.137698397764&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.004697758905&{}:{}&0.418624066210&{}:{}&0.586073692694&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.297168040796&{}:{}&0.009439864014&{}:{}&0.693392095190&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.027539679553&{}:{}&0.045899465921&{}:{}&0.926560854526&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1}}&{}\approx{}&0.025384292556&{}:{}&0.043935242169&{}:{}&0.930680465275&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.050407872089&{}:{}&0.042807190345&{}:{}&0.906784937566&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.024354305496&{}:{}&0.082728299997&{}:{}&0.892917394507&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.133575265356&{}:{}&0.231192640811&{}:{}&0.635232093834&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1}}&{}\approx{}&-0.000555688782&{}:{}&0.000943800024&{}:{}&0.999611888757&,\\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.028838104788&{}:{}&0.000916073569&{}:{}&0.970245821643&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.000529946914&{}:{}&0.047224333205&{}:{}&0.953305613708&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.007971020534&{}:{}&0.013538242306&{}:{}&0.994432778228&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1}}&{}\approx{}&0.045382591818&{}:{}&0.077079280111&{}:{}&0.877538128072&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.004697758905&{}:{}&0.418624066210&{}:{}&0.586073692694&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.297168040796&{}:{}&0.009439864014&{}:{}&0.693392095190&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.027539679553&{}:{}&0.045899465921&{}:{}&0.926560854526&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.050407872089&{}:{}&0.042807190345&{}:{}&0.906784937566&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.024354305496&{}:{}&0.082728299997&{}:{}&0.892917394507&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.133575265356&{}:{}&0.231192640811&{}:{}&0.635232093834&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1}}&{}\approx{}&-0.064934692603&{}:{}&-0.114613828991&{}:{}&1.179548521594&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.107625218511&{}:{}&1.107625218511&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.058257569496&{}:{}&0.000000000000&{}:{}&1.058257569496&,\\C^*_{\mathbf{1}}&{}\approx{}&0.361655401151&{}:{}&0.638344598849&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1}}&{}\approx{}&0.083698291999&{}:{}&0.142155920044&{}:{}&0.774145787956&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.050407872089&{}:{}&0.042807190345&{}:{}&0.906784937566&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.024354305496&{}:{}&0.082728299997&{}:{}&0.892917394507&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.133575265356&{}:{}&0.231192640811&{}:{}&0.635232093834&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.107625218511&{}:{}&1.107625218511&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.058257569496&{}:{}&0.000000000000&{}:{}&1.058257569496&,\\C^*_{\mathbf{1}}&{}\approx{}&0.361655401151&{}:{}&0.638344598849&{}:{}&0.000000000000&. \end{alignedat} \]
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