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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4(200)

Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&, \\ P_{\mathbf{4}}&{}\approx{}&0.989795649994&{}:{}&0.005102175003&{}:{}&0.005102175003&, \\ P^-_{\mathbf{4}}&{}\approx{}&1.029567858583&{}:{}&-0.014783929292&{}:{}&-0.014783929292&, \\ P^+_{\mathbf{4}}&{}\approx{}&0.954321849295&{}:{}&0.022839075352&{}:{}&0.022839075352&, \\ Q_{\mathbf{4}}&{}\approx{}&1.167669846111&{}:{}&-0.083834923055&{}:{}&-0.083834923055&, \\ I^\prime_{\mathbf{4}}&{}\approx{}&0.874436559411&{}:{}&0.062781720295&{}:{}&0.062781720295&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{4}}\)
\(P^-_{\mathbf{4}}\)
\(P^+_{\mathbf{4}}\)
\(Q_{\mathbf{4}}\)
\(I^\prime_{\mathbf{4}}\)
4 (200)

Central Triangles

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

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

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4}}&{}\approx{}&0.989795649994&{}:{}&0.005102175003&{}:{}&0.005102175003&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.788675134595&{}:{}&0.105662432703&{}:{}&0.105662432703&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.958193491830&{}:{}&0.036867235218&{}:{}&0.004939272952&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.958193491830&{}:{}&0.004939272952&{}:{}&0.036867235218&. \end{alignedat} \]
4 (200)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4}}&{}\approx{}&1.029567858583&{}:{}&-0.014783929292&{}:{}&-0.014783929292&,\\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-2.154700538379&{}:{}&1.577350269190&{}:{}&1.577350269190&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.994771591605&{}:{}&0.019512685773&{}:{}&-0.014284277378&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.994771591605&{}:{}&-0.014284277378&{}:{}&0.019512685773&. \end{alignedat} \]
4 (200)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4}}&{}\approx{}&0.954321849295&{}:{}&0.022839075352&{}:{}&0.022839075352&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.962250448649&{}:{}&0.018874775675&{}:{}&0.018874775675&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.788675134595&{}:{}&-0.577350269190&{}:{}&0.788675134595&,\\C^\prime_{\mathbf{4}}&{}\approx{}&0.788675134595&{}:{}&0.788675134595&{}:{}&-0.577350269190&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.788675134595&{}:{}&0.105662432703&{}:{}&0.105662432703&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.958193491830&{}:{}&0.036867235218&{}:{}&0.004939272952&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.958193491830&{}:{}&0.004939272952&{}:{}&0.036867235218&, \end{alignedat} \]
4 (200)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4}}&{}\approx{}&1.167669846111&{}:{}&-0.083834923055&{}:{}&-0.083834923055&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.500000000000&{}:{}&0.500000000000&,\\B^*_{\mathbf{4}}&{}\approx{}&1.077350269190&{}:{}&0.000000000000&{}:{}&-0.077350269190&,\\C^*_{\mathbf{4}}&{}\approx{}&1.077350269190&{}:{}&-0.077350269190&{}:{}&0.000000000000&. \end{alignedat} \]
4 (200)

Radical center of the Malfatti circles

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