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,
\[ \begin{alignedat}{4} I&{}\approx{}&0.267949192431&{}:{}&0.267949192431&{}:{}&0.464101615138&, \\ P_{\mathbf{1}}&{}\approx{}&0.034750498383&{}:{}&0.034750498383&{}:{}&0.930499003235&, \\ P^-_{\mathbf{1}}&{}\approx{}&-0.000000000000&{}:{}&-0.000000000000&{}:{}&1.000000000000&, \\ P^+_{\mathbf{1}}&{}\approx{}&0.061522150572&{}:{}&0.061522150572&{}:{}&0.876955698855&, \\ Q_{\mathbf{1}}&{}\approx{}&-0.089635998603&{}:{}&-0.089635998603&{}:{}&1.179271997205&, \\ I^\prime_{\mathbf{1}}&{}\approx{}&0.113505228594&{}:{}&0.113505228594&{}:{}&0.772989542812&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{1}}\)
\(P^-_{\mathbf{1}}\)
\(P^+_{\mathbf{1}}\)
\(Q_{\mathbf{1}}\)
\(I^\prime_{\mathbf{1}}\)
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1}}&{}\approx{}&-0.000000000000&{}:{}&0.366025403784&{}:{}&0.633974596216&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.366025403784&{}:{}&-0.000000000000&{}:{}&0.633974596216&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.036976074586&{}:{}&0.036976074586&{}:{}&0.926047850828&, \\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.067166913062&{}:{}&0.033583456531&{}:{}&0.899249630407&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.033583456531&{}:{}&0.067166913062&{}:{}&0.899249630407&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.183012701892&{}:{}&0.183012701892&{}:{}&0.633974596216&, \\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.038395800572&{}:{}&-0.000000000000&{}:{}&0.961604199428&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.000000000000&{}:{}&0.038395800572&{}:{}&0.961604199428&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.000000000000&{}:{}&-0.000000000000&{}:{}&1.000000000000&, \\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.082262332300&{}:{}&1.082262332300&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.082262332300&{}:{}&0.000000000000&{}:{}&1.082262332300&,\\C^*_{\mathbf{1}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1}}}{B^\prime_{\mathbf{1}}}{C^\prime_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1}}}{B^{\prime\prime}_{\mathbf{1}}}{C^{\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1}}}{B^{\prime\prime\prime}_{\mathbf{1}}}{C^{\prime\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^*_{\mathbf{1}}}{B^*_{\mathbf{1}}}{C^*_{\mathbf{1}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1}}}}&{}\approx{}&1.366025403784&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1}}}}&{}\approx{}&1.366025403784&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1}}}}&{}\approx{}&0.137996589020&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.267949192431&{}:{}&0.267949192431&{}:{}&0.464101615138&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.000000000000&{}:{}&0.366025403784&{}:{}&0.633974596216&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.366025403784&{}:{}&-0.000000000000&{}:{}&0.633974596216&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.036976074586&{}:{}&0.036976074586&{}:{}&0.926047850828&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1}}&{}\approx{}&0.034750498383&{}:{}&0.034750498383&{}:{}&0.930499003235&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.067166913062&{}:{}&0.033583456531&{}:{}&0.899249630407&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.033583456531&{}:{}&0.067166913062&{}:{}&0.899249630407&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.183012701892&{}:{}&0.183012701892&{}:{}&0.633974596216&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1}}&{}\approx{}&-0.000000000000&{}:{}&-0.000000000000&{}:{}&1.000000000000&,\\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.038395800572&{}:{}&-0.000000000000&{}:{}&0.961604199428&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.000000000000&{}:{}&0.038395800572&{}:{}&0.961604199428&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.000000000000&{}:{}&-0.000000000000&{}:{}&1.000000000000&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1}}&{}\approx{}&0.061522150572&{}:{}&0.061522150572&{}:{}&0.876955698855&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.000000000000&{}:{}&0.366025403784&{}:{}&0.633974596216&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.366025403784&{}:{}&-0.000000000000&{}:{}&0.633974596216&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.036976074586&{}:{}&0.036976074586&{}:{}&0.926047850828&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.067166913062&{}:{}&0.033583456531&{}:{}&0.899249630407&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.033583456531&{}:{}&0.067166913062&{}:{}&0.899249630407&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.183012701892&{}:{}&0.183012701892&{}:{}&0.633974596216&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1}}&{}\approx{}&-0.089635998603&{}:{}&-0.089635998603&{}:{}&1.179271997205&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.082262332300&{}:{}&1.082262332300&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.082262332300&{}:{}&0.000000000000&{}:{}&1.082262332300&,\\C^*_{\mathbf{1}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1}}&{}\approx{}&0.113505228594&{}:{}&0.113505228594&{}:{}&0.772989542812&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.067166913062&{}:{}&0.033583456531&{}:{}&0.899249630407&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.033583456531&{}:{}&0.067166913062&{}:{}&0.899249630407&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.183012701892&{}:{}&0.183012701892&{}:{}&0.633974596216&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.082262332300&{}:{}&1.082262332300&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.082262332300&{}:{}&0.000000000000&{}:{}&1.082262332300&,\\C^*_{\mathbf{1}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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