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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5a(213)

Malfatti circles

5a (213)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-1.000000000000&{}:{}&1.000000000000&{}:{}&1.000000000000&, \\ P_{\mathbf{5a}}&{}\approx{}&1.086689796779&{}:{}&-0.008668979678&{}:{}&-0.078020817101&, \\ P^-_{\mathbf{5a}}&{}\approx{}&0.902075806737&{}:{}&0.080570161309&{}:{}&0.017354031954&, \\ P^+_{\mathbf{5a}}&{}\approx{}&1.310992966503&{}:{}&-0.117093168886&{}:{}&-0.193899797618&, \\ Q_{\mathbf{5a}}&{}\approx{}&1.448018475480&{}:{}&0.224009237740&{}:{}&-0.672027713219&, \\ I^\prime_{\mathbf{5a}}&{}\approx{}&1.555852595253&{}:{}&-0.138963148813&{}:{}&-0.416889446440&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{5a}}\)
\(P^-_{\mathbf{5a}}\)
\(P^+_{\mathbf{5a}}\)
\(Q_{\mathbf{5a}}\)
\(I^\prime_{\mathbf{5a}}\)
5a (213)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5a}}&{}\approx{}&1.244016935856&{}:{}&-0.122008467928&{}:{}&-0.122008467928&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&1.098076211353&{}:{}&1.000000000000&{}:{}&-1.098076211353&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&5.098076211353&{}:{}&-5.098076211353&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.806952388982&{}:{}&-0.080695238898&{}:{}&-0.726257150084&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.334097441518&{}:{}&-0.238313554719&{}:{}&-0.095783886799&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.229422863406&{}:{}&-0.009807621135&{}:{}&-0.219615242271&, \\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.474645386743&{}:{}&0.432251770720&{}:{}&0.093102842536&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.085585763082&{}:{}&-0.106470138742&{}:{}&0.020884375660&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.008837226334&{}:{}&0.090105706697&{}:{}&-0.098942933031&, \\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.500000000000&{}:{}&1.500000000000&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.866025403784&{}:{}&0.000000000000&{}:{}&-0.866025403784&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.866025403784&{}:{}&0.133974596216&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5a}}}{B^\prime_{\mathbf{5a}}}{C^\prime_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^*_{\mathbf{5a}}}{B^*_{\mathbf{5a}}}{C^*_{\mathbf{5a}}}\)
5a (213)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5a}}}}&{}\approx{}&-0.122008467928&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5a}}}}&{}\approx{}&-1.098076211353&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5a}}}}&{}\approx{}&-5.098076211353&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-1.000000000000&{}:{}&1.000000000000&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.244016935856&{}:{}&-0.122008467928&{}:{}&-0.122008467928&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&1.098076211353&{}:{}&1.000000000000&{}:{}&-1.098076211353&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&5.098076211353&{}:{}&-5.098076211353&{}:{}&1.000000000000&. \end{alignedat} \]
5a (213)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5a}}&{}\approx{}&1.086689796779&{}:{}&-0.008668979678&{}:{}&-0.078020817101&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.806952388982&{}:{}&-0.080695238898&{}:{}&-0.726257150084&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.334097441518&{}:{}&-0.238313554719&{}:{}&-0.095783886799&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.229422863406&{}:{}&-0.009807621135&{}:{}&-0.219615242271&. \end{alignedat} \]
5a (213)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5a}}&{}\approx{}&0.902075806737&{}:{}&0.080570161309&{}:{}&0.017354031954&,\\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.474645386743&{}:{}&0.432251770720&{}:{}&0.093102842536&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.085585763082&{}:{}&-0.106470138742&{}:{}&0.020884375660&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.008837226334&{}:{}&0.090105706697&{}:{}&-0.098942933031&. \end{alignedat} \]
5a (213)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5a}}&{}\approx{}&1.310992966503&{}:{}&-0.117093168886&{}:{}&-0.193899797618&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.244016935856&{}:{}&-0.122008467928&{}:{}&-0.122008467928&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&1.098076211353&{}:{}&1.000000000000&{}:{}&-1.098076211353&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&5.098076211353&{}:{}&-5.098076211353&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.806952388982&{}:{}&-0.080695238898&{}:{}&-0.726257150084&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.334097441518&{}:{}&-0.238313554719&{}:{}&-0.095783886799&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.229422863406&{}:{}&-0.009807621135&{}:{}&-0.219615242271&, \end{alignedat} \]
5a (213)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5a}}&{}\approx{}&1.448018475480&{}:{}&0.224009237740&{}:{}&-0.672027713219&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.500000000000&{}:{}&1.500000000000&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.866025403784&{}:{}&0.000000000000&{}:{}&-0.866025403784&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.866025403784&{}:{}&0.133974596216&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5a}}&{}\approx{}&1.555852595253&{}:{}&-0.138963148813&{}:{}&-0.416889446440&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.806952388982&{}:{}&-0.080695238898&{}:{}&-0.726257150084&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.334097441518&{}:{}&-0.238313554719&{}:{}&-0.095783886799&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.229422863406&{}:{}&-0.009807621135&{}:{}&-0.219615242271&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.500000000000&{}:{}&1.500000000000&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.866025403784&{}:{}&0.000000000000&{}:{}&-0.866025403784&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.866025403784&{}:{}&0.133974596216&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)