Derousseau's Generalization of the Malfatti circles

Equilateral Triangle

\(a:b:c=1:1:1\), \(A=60\degree\), \(B=60\degree\), \(C=60\degree\).


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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&, \\ P_{\mathbf{1}}&{}\approx{}&0.005102175003&{}:{}&0.005102175003&{}:{}&0.989795649994&, \\ P^-_{\mathbf{1}}&{}\approx{}&-0.014783929292&{}:{}&-0.014783929292&{}:{}&1.029567858583&, \\ P^+_{\mathbf{1}}&{}\approx{}&0.022839075352&{}:{}&0.022839075352&{}:{}&0.954321849295&, \\ Q_{\mathbf{1}}&{}\approx{}&-0.083834923055&{}:{}&-0.083834923055&{}:{}&1.167669846111&, \\ I^\prime_{\mathbf{1}}&{}\approx{}&0.062781720295&{}:{}&0.062781720295&{}:{}&0.874436559411&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{1}}\)
\(P^-_{\mathbf{1}}\)
\(P^+_{\mathbf{1}}\)
\(Q_{\mathbf{1}}\)
\(I^\prime_{\mathbf{1}}\)
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1}}&{}\approx{}&-0.577350269190&{}:{}&0.788675134595&{}:{}&0.788675134595&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.788675134595&{}:{}&-0.577350269190&{}:{}&0.788675134595&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.018874775675&{}:{}&0.018874775675&{}:{}&0.962250448649&, \\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.036867235218&{}:{}&0.004939272952&{}:{}&0.958193491830&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.004939272952&{}:{}&0.036867235218&{}:{}&0.958193491830&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.105662432703&{}:{}&0.105662432703&{}:{}&0.788675134595&, \\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.019512685773&{}:{}&-0.014284277378&{}:{}&0.994771591605&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.014284277378&{}:{}&0.019512685773&{}:{}&0.994771591605&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&1.577350269190&{}:{}&1.577350269190&{}:{}&-2.154700538379&, \\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.077350269190&{}:{}&1.077350269190&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.077350269190&{}:{}&0.000000000000&{}:{}&1.077350269190&,\\C^*_{\mathbf{1}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1}}}{B^\prime_{\mathbf{1}}}{C^\prime_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1}}}{B^{\prime\prime}_{\mathbf{1}}}{C^{\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1}}}{B^{\prime\prime\prime}_{\mathbf{1}}}{C^{\prime\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^*_{\mathbf{1}}}{B^*_{\mathbf{1}}}{C^*_{\mathbf{1}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1}}}}&{}\approx{}&2.366025403784&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1}}}}&{}\approx{}&2.366025403784&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1}}}}&{}\approx{}&0.056624327026&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.577350269190&{}:{}&0.788675134595&{}:{}&0.788675134595&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.788675134595&{}:{}&-0.577350269190&{}:{}&0.788675134595&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.018874775675&{}:{}&0.018874775675&{}:{}&0.962250448649&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1}}&{}\approx{}&0.005102175003&{}:{}&0.005102175003&{}:{}&0.989795649994&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.036867235218&{}:{}&0.004939272952&{}:{}&0.958193491830&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.004939272952&{}:{}&0.036867235218&{}:{}&0.958193491830&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.105662432703&{}:{}&0.105662432703&{}:{}&0.788675134595&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1}}&{}\approx{}&-0.014783929292&{}:{}&-0.014783929292&{}:{}&1.029567858583&,\\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.019512685773&{}:{}&-0.014284277378&{}:{}&0.994771591605&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.014284277378&{}:{}&0.019512685773&{}:{}&0.994771591605&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&1.577350269190&{}:{}&1.577350269190&{}:{}&-2.154700538379&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1}}&{}\approx{}&0.022839075352&{}:{}&0.022839075352&{}:{}&0.954321849295&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.577350269190&{}:{}&0.788675134595&{}:{}&0.788675134595&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.788675134595&{}:{}&-0.577350269190&{}:{}&0.788675134595&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.018874775675&{}:{}&0.018874775675&{}:{}&0.962250448649&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.036867235218&{}:{}&0.004939272952&{}:{}&0.958193491830&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.004939272952&{}:{}&0.036867235218&{}:{}&0.958193491830&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.105662432703&{}:{}&0.105662432703&{}:{}&0.788675134595&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1}}&{}\approx{}&-0.083834923055&{}:{}&-0.083834923055&{}:{}&1.167669846111&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.077350269190&{}:{}&1.077350269190&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.077350269190&{}:{}&0.000000000000&{}:{}&1.077350269190&,\\C^*_{\mathbf{1}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1}}&{}\approx{}&0.062781720295&{}:{}&0.062781720295&{}:{}&0.874436559411&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.036867235218&{}:{}&0.004939272952&{}:{}&0.958193491830&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.004939272952&{}:{}&0.036867235218&{}:{}&0.958193491830&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.105662432703&{}:{}&0.105662432703&{}:{}&0.788675134595&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.077350269190&{}:{}&1.077350269190&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.077350269190&{}:{}&0.000000000000&{}:{}&1.077350269190&,\\C^*_{\mathbf{1}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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