Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 120° Top Angle

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

[Top] > Isosceles Triangle with 120° Top Angle > 4 (200)

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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.267949192431&{}:{}&0.267949192431&{}:{}&0.464101615138&, \\ P_{\mathbf{4}}&{}\approx{}&0.998807236804&{}:{}&0.000300053081&{}:{}&0.000892710115&, \\ P^-_{\mathbf{4}}&{}\approx{}&1.014893470850&{}:{}&-0.005590922234&{}:{}&-0.009302548617&, \\ P^+_{\mathbf{4}}&{}\approx{}&0.983399263774&{}:{}&0.005942640930&{}:{}&0.010658095296&, \\ Q_{\mathbf{4}}&{}\approx{}&1.102951701196&{}:{}&-0.019116778140&{}:{}&-0.083834923055&, \\ I^\prime_{\mathbf{4}}&{}\approx{}&0.946362226653&{}:{}&0.016402709845&{}:{}&0.037235063502&, \end{alignedat} \]
\(I\) Incenter
\(P_{\mathbf{4}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{4}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{4}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{4}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{4}}\) Radical center of the Malfatti circles
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4}}&{}\approx{}&0.984372072030&{}:{}&0.005720218646&{}:{}&0.009907709325&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.560523051614&{}:{}&-0.531377455822&{}:{}&0.970854404209&,\\C^\prime_{\mathbf{4}}&{}\approx{}&2.366025403784&{}:{}&2.366025403784&{}:{}&-3.732050807569&, \\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.906331673462&{}:{}&0.023563327620&{}:{}&0.070104998917&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.987704368325&{}:{}&0.011412845039&{}:{}&0.000882786636&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.980090327421&{}:{}&0.000294430308&{}:{}&0.019615242271&, \\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-0.655585897522&{}:{}&0.621497305589&{}:{}&1.034088591933&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&1.003366295423&{}:{}&0.005830594706&{}:{}&-0.009196890128&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.995464535619&{}:{}&-0.005483890640&{}:{}&0.010019355020&, \\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.185686860132&{}:{}&0.814313139868&,\\B^*_{\mathbf{4}}&{}\approx{}&1.082262332300&{}:{}&0.000000000000&{}:{}&-0.082262332300&,\\C^*_{\mathbf{4}}&{}\approx{}&1.017638090205&{}:{}&-0.017638090205&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4}}}{B^\prime_{\mathbf{4}}}{C^\prime_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4}}}{B^{\prime\prime}_{\mathbf{4}}}{C^{\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4}}}{B^{\prime\prime\prime}_{\mathbf{4}}}{C^{\prime\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^*_{\mathbf{4}}}{B^*_{\mathbf{4}}}{C^*_{\mathbf{4}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4}}}}&{}\approx{}&0.021348146616&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4}}}}&{}\approx{}&2.091900507436&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4}}}}&{}\approx{}&8.830127018922&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.267949192431&{}:{}&0.267949192431&{}:{}&0.464101615138&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.984372072030&{}:{}&0.005720218646&{}:{}&0.009907709325&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.560523051614&{}:{}&-0.531377455822&{}:{}&0.970854404209&,\\C^\prime_{\mathbf{4}}&{}\approx{}&2.366025403784&{}:{}&2.366025403784&{}:{}&-3.732050807569&. \end{alignedat} \]
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First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4}}&{}\approx{}&0.998807236804&{}:{}&0.000300053081&{}:{}&0.000892710115&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.906331673462&{}:{}&0.023563327620&{}:{}&0.070104998917&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.987704368325&{}:{}&0.011412845039&{}:{}&0.000882786636&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.980090327421&{}:{}&0.000294430308&{}:{}&0.019615242271&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4}}&{}\approx{}&1.014893470850&{}:{}&-0.005590922234&{}:{}&-0.009302548617&,\\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-0.655585897522&{}:{}&0.621497305589&{}:{}&1.034088591933&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&1.003366295423&{}:{}&0.005830594706&{}:{}&-0.009196890128&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.995464535619&{}:{}&-0.005483890640&{}:{}&0.010019355020&. \end{alignedat} \]
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Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4}}&{}\approx{}&0.983399263774&{}:{}&0.005942640930&{}:{}&0.010658095296&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.984372072030&{}:{}&0.005720218646&{}:{}&0.009907709325&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.560523051614&{}:{}&-0.531377455822&{}:{}&0.970854404209&,\\C^\prime_{\mathbf{4}}&{}\approx{}&2.366025403784&{}:{}&2.366025403784&{}:{}&-3.732050807569&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.906331673462&{}:{}&0.023563327620&{}:{}&0.070104998917&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.987704368325&{}:{}&0.011412845039&{}:{}&0.000882786636&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.980090327421&{}:{}&0.000294430308&{}:{}&0.019615242271&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4}}&{}\approx{}&1.102951701196&{}:{}&-0.019116778140&{}:{}&-0.083834923055&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.185686860132&{}:{}&0.814313139868&,\\B^*_{\mathbf{4}}&{}\approx{}&1.082262332300&{}:{}&0.000000000000&{}:{}&-0.082262332300&,\\C^*_{\mathbf{4}}&{}\approx{}&1.017638090205&{}:{}&-0.017638090205&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4}}&{}\approx{}&0.946362226653&{}:{}&0.016402709845&{}:{}&0.037235063502&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.906331673462&{}:{}&0.023563327620&{}:{}&0.070104998917&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.987704368325&{}:{}&0.011412845039&{}:{}&0.000882786636&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.980090327421&{}:{}&0.000294430308&{}:{}&0.019615242271&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.185686860132&{}:{}&0.814313139868&,\\B^*_{\mathbf{4}}&{}\approx{}&1.082262332300&{}:{}&0.000000000000&{}:{}&-0.082262332300&,\\C^*_{\mathbf{4}}&{}\approx{}&1.017638090205&{}:{}&-0.017638090205&{}:{}&0.000000000000&. \end{alignedat} \]
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