Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 120° Top Angle

\(A=30\degree\), \(B=30\degree\), \(C=120\degree\).

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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.267949192431&{}:{}&0.267949192431&{}:{}&0.464101615138&, \\ P_{\mathbf{3}}&{}\approx{}&0.000297925393&{}:{}&0.991724655749&{}:{}&0.007977418858&, \\ P^-_{\mathbf{3}}&{}\approx{}&-0.018048858103&{}:{}&1.041337541472&{}:{}&-0.023288683369&, \\ P^+_{\mathbf{3}}&{}\approx{}&0.016432714946&{}:{}&0.948093385425&{}:{}&0.035473899628&, \\ Q_{\mathbf{3}}&{}\approx{}&-0.014316041268&{}:{}&0.825970880384&{}:{}&0.188345160884&, \\ I^\prime_{\mathbf{3}}&{}\approx{}&0.015265859359&{}:{}&0.880771091600&{}:{}&0.103963049041&, \end{alignedat} \]
\(I\) Incenter
\(P_{\mathbf{3}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{3}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{3}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{3}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{3}}\) Radical center of the Malfatti circles
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3}}&{}\approx{}&-0.561722341219&{}:{}&0.571630050544&{}:{}&0.990092290675&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.016827217576&{}:{}&0.954027186633&{}:{}&0.029145595791&,\\C^\prime_{\mathbf{3}}&{}\approx{}&1.366025403784&{}:{}&1.366025403784&{}:{}&-1.732050807569&, \\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.033244911755&{}:{}&0.959040579625&{}:{}&0.007714508620&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006984824160&{}:{}&0.805985572786&{}:{}&0.187029603054&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000283315696&{}:{}&0.943092357278&{}:{}&0.056624327026&, \\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.017721845633&{}:{}&1.004748554226&{}:{}&-0.022470399859&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.785746369477&{}:{}&-0.799605434820&{}:{}&1.013859065343&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.017106274714&{}:{}&0.986954739918&{}:{}&0.030151534796&, \\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.814313139868&{}:{}&0.185686860132&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.082262332300&{}:{}&0.000000000000&{}:{}&1.082262332300&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.017638090205&{}:{}&1.017638090205&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3}}}{B^\prime_{\mathbf{3}}}{C^\prime_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3}}}{B^{\prime\prime}_{\mathbf{3}}}{C^{\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3}}}{B^{\prime\prime\prime}_{\mathbf{3}}}{C^{\prime\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^*_{\mathbf{3}}}{B^*_{\mathbf{3}}}{C^*_{\mathbf{3}}}\)
3 (022)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3}}}}&{}\approx{}&2.133352391763&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3}}}}&{}\approx{}&0.062800030943&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3}}}}&{}\approx{}&5.098076211353&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.267949192431&{}:{}&0.267949192431&{}:{}&0.464101615138&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.561722341219&{}:{}&0.571630050544&{}:{}&0.990092290675&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.016827217576&{}:{}&0.954027186633&{}:{}&0.029145595791&,\\C^\prime_{\mathbf{3}}&{}\approx{}&1.366025403784&{}:{}&1.366025403784&{}:{}&-1.732050807569&. \end{alignedat} \]
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First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3}}&{}\approx{}&0.000297925393&{}:{}&0.991724655749&{}:{}&0.007977418858&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.033244911755&{}:{}&0.959040579625&{}:{}&0.007714508620&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006984824160&{}:{}&0.805985572786&{}:{}&0.187029603054&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000283315696&{}:{}&0.943092357278&{}:{}&0.056624327026&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3}}&{}\approx{}&-0.018048858103&{}:{}&1.041337541472&{}:{}&-0.023288683369&,\\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.017721845633&{}:{}&1.004748554226&{}:{}&-0.022470399859&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.785746369477&{}:{}&-0.799605434820&{}:{}&1.013859065343&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.017106274714&{}:{}&0.986954739918&{}:{}&0.030151534796&. \end{alignedat} \]
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Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3}}&{}\approx{}&0.016432714946&{}:{}&0.948093385425&{}:{}&0.035473899628&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.561722341219&{}:{}&0.571630050544&{}:{}&0.990092290675&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.016827217576&{}:{}&0.954027186633&{}:{}&0.029145595791&,\\C^\prime_{\mathbf{3}}&{}\approx{}&1.366025403784&{}:{}&1.366025403784&{}:{}&-1.732050807569&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.033244911755&{}:{}&0.959040579625&{}:{}&0.007714508620&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006984824160&{}:{}&0.805985572786&{}:{}&0.187029603054&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000283315696&{}:{}&0.943092357278&{}:{}&0.056624327026&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3}}&{}\approx{}&-0.014316041268&{}:{}&0.825970880384&{}:{}&0.188345160884&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.814313139868&{}:{}&0.185686860132&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.082262332300&{}:{}&0.000000000000&{}:{}&1.082262332300&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.017638090205&{}:{}&1.017638090205&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3}}&{}\approx{}&0.015265859359&{}:{}&0.880771091600&{}:{}&0.103963049041&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.033244911755&{}:{}&0.959040579625&{}:{}&0.007714508620&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006984824160&{}:{}&0.805985572786&{}:{}&0.187029603054&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000283315696&{}:{}&0.943092357278&{}:{}&0.056624327026&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.814313139868&{}:{}&0.185686860132&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.082262332300&{}:{}&0.000000000000&{}:{}&1.082262332300&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.017638090205&{}:{}&1.017638090205&{}:{}&0.000000000000&. \end{alignedat} \]
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