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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6a(231)

Malfatti circles

6a (231)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-1.000000000000&{}:{}&1.000000000000&{}:{}&1.000000000000&, \\ P_{\mathbf{6a}}&{}\approx{}&1.086689796779&{}:{}&-0.078020817101&{}:{}&-0.008668979678&, \\ P^-_{\mathbf{6a}}&{}\approx{}&0.902075806737&{}:{}&0.017354031954&{}:{}&0.080570161309&, \\ P^+_{\mathbf{6a}}&{}\approx{}&1.310992966503&{}:{}&-0.193899797618&{}:{}&-0.117093168886&, \\ Q_{\mathbf{6a}}&{}\approx{}&1.448018475480&{}:{}&-0.672027713219&{}:{}&0.224009237740&, \\ I^\prime_{\mathbf{6a}}&{}\approx{}&1.555852595253&{}:{}&-0.416889446440&{}:{}&-0.138963148813&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{6a}}\)
\(P^-_{\mathbf{6a}}\)
\(P^+_{\mathbf{6a}}\)
\(Q_{\mathbf{6a}}\)
\(I^\prime_{\mathbf{6a}}\)
6a (231)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6a}}&{}\approx{}&1.244016935856&{}:{}&-0.122008467928&{}:{}&-0.122008467928&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&5.098076211353&{}:{}&1.000000000000&{}:{}&-5.098076211353&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&1.098076211353&{}:{}&-1.098076211353&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.806952388982&{}:{}&-0.726257150084&{}:{}&-0.080695238898&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.229422863406&{}:{}&-0.219615242271&{}:{}&-0.009807621135&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.334097441518&{}:{}&-0.095783886799&{}:{}&-0.238313554719&, \\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.474645386743&{}:{}&0.093102842536&{}:{}&0.432251770720&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.008837226334&{}:{}&-0.098942933031&{}:{}&0.090105706697&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.085585763082&{}:{}&0.020884375660&{}:{}&-0.106470138742&, \\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.500000000000&{}:{}&-0.500000000000&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.866025403784&{}:{}&0.000000000000&{}:{}&0.133974596216&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.866025403784&{}:{}&-0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6a}}}{B^\prime_{\mathbf{6a}}}{C^\prime_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^*_{\mathbf{6a}}}{B^*_{\mathbf{6a}}}{C^*_{\mathbf{6a}}}\)
6a (231)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6a}}}}&{}\approx{}&-0.122008467928&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6a}}}}&{}\approx{}&-5.098076211353&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6a}}}}&{}\approx{}&-1.098076211353&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-1.000000000000&{}:{}&1.000000000000&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.244016935856&{}:{}&-0.122008467928&{}:{}&-0.122008467928&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&5.098076211353&{}:{}&1.000000000000&{}:{}&-5.098076211353&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&1.098076211353&{}:{}&-1.098076211353&{}:{}&1.000000000000&. \end{alignedat} \]
6a (231)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6a}}&{}\approx{}&1.086689796779&{}:{}&-0.078020817101&{}:{}&-0.008668979678&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.806952388982&{}:{}&-0.726257150084&{}:{}&-0.080695238898&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.229422863406&{}:{}&-0.219615242271&{}:{}&-0.009807621135&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.334097441518&{}:{}&-0.095783886799&{}:{}&-0.238313554719&. \end{alignedat} \]
6a (231)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6a}}&{}\approx{}&0.902075806737&{}:{}&0.017354031954&{}:{}&0.080570161309&,\\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.474645386743&{}:{}&0.093102842536&{}:{}&0.432251770720&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.008837226334&{}:{}&-0.098942933031&{}:{}&0.090105706697&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.085585763082&{}:{}&0.020884375660&{}:{}&-0.106470138742&. \end{alignedat} \]
6a (231)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6a}}&{}\approx{}&1.310992966503&{}:{}&-0.193899797618&{}:{}&-0.117093168886&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.244016935856&{}:{}&-0.122008467928&{}:{}&-0.122008467928&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&5.098076211353&{}:{}&1.000000000000&{}:{}&-5.098076211353&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&1.098076211353&{}:{}&-1.098076211353&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.806952388982&{}:{}&-0.726257150084&{}:{}&-0.080695238898&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.229422863406&{}:{}&-0.219615242271&{}:{}&-0.009807621135&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.334097441518&{}:{}&-0.095783886799&{}:{}&-0.238313554719&, \end{alignedat} \]
6a (231)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6a}}&{}\approx{}&1.448018475480&{}:{}&-0.672027713219&{}:{}&0.224009237740&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.500000000000&{}:{}&-0.500000000000&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.866025403784&{}:{}&0.000000000000&{}:{}&0.133974596216&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.866025403784&{}:{}&-0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6a}}&{}\approx{}&1.555852595253&{}:{}&-0.416889446440&{}:{}&-0.138963148813&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.806952388982&{}:{}&-0.726257150084&{}:{}&-0.080695238898&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.229422863406&{}:{}&-0.219615242271&{}:{}&-0.009807621135&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.334097441518&{}:{}&-0.095783886799&{}:{}&-0.238313554719&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.500000000000&{}:{}&-0.500000000000&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.866025403784&{}:{}&0.000000000000&{}:{}&0.133974596216&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.866025403784&{}:{}&-0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)