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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3b(123)

Malfatti circles

3b (123)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&1.000000000000&{}:{}&-1.000000000000&{}:{}&1.000000000000&, \\ P_{\mathbf{3b}}&{}\approx{}&-0.008668979678&{}:{}&1.086689796779&{}:{}&-0.078020817101&, \\ P^-_{\mathbf{3b}}&{}\approx{}&0.080570161309&{}:{}&0.902075806737&{}:{}&0.017354031954&, \\ P^+_{\mathbf{3b}}&{}\approx{}&-0.117093168886&{}:{}&1.310992966503&{}:{}&-0.193899797618&, \\ Q_{\mathbf{3b}}&{}\approx{}&0.224009237740&{}:{}&1.448018475480&{}:{}&-0.672027713219&, \\ I^\prime_{\mathbf{3b}}&{}\approx{}&-0.138963148813&{}:{}&1.555852595253&{}:{}&-0.416889446440&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{3b}}\)
\(P^-_{\mathbf{3b}}\)
\(P^+_{\mathbf{3b}}\)
\(Q_{\mathbf{3b}}\)
\(I^\prime_{\mathbf{3b}}\)
3b (123)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3b}}&{}\approx{}&1.000000000000&{}:{}&1.098076211353&{}:{}&-1.098076211353&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.122008467928&{}:{}&1.244016935856&{}:{}&-0.122008467928&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-5.098076211353&{}:{}&5.098076211353&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.238313554719&{}:{}&1.334097441518&{}:{}&-0.095783886799&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.080695238898&{}:{}&1.806952388982&{}:{}&-0.726257150084&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.009807621135&{}:{}&1.229422863406&{}:{}&-0.219615242271&, \\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.106470138742&{}:{}&1.085585763082&{}:{}&0.020884375660&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.432251770720&{}:{}&0.474645386743&{}:{}&0.093102842536&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.090105706697&{}:{}&1.008837226334&{}:{}&-0.098942933031&, \\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&1.866025403784&{}:{}&-0.866025403784&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.500000000000&{}:{}&0.000000000000&{}:{}&1.500000000000&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.133974596216&{}:{}&0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3b}}}{B^\prime_{\mathbf{3b}}}{C^\prime_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^*_{\mathbf{3b}}}{B^*_{\mathbf{3b}}}{C^*_{\mathbf{3b}}}\)
3b (123)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3b}}}}&{}\approx{}&-1.098076211353&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3b}}}}&{}\approx{}&-0.122008467928&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3b}}}}&{}\approx{}&-5.098076211353&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&1.000000000000&{}:{}&-1.000000000000&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&1.000000000000&{}:{}&1.098076211353&{}:{}&-1.098076211353&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.122008467928&{}:{}&1.244016935856&{}:{}&-0.122008467928&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-5.098076211353&{}:{}&5.098076211353&{}:{}&1.000000000000&. \end{alignedat} \]
3b (123)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3b}}&{}\approx{}&-0.008668979678&{}:{}&1.086689796779&{}:{}&-0.078020817101&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.238313554719&{}:{}&1.334097441518&{}:{}&-0.095783886799&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.080695238898&{}:{}&1.806952388982&{}:{}&-0.726257150084&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.009807621135&{}:{}&1.229422863406&{}:{}&-0.219615242271&. \end{alignedat} \]
3b (123)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3b}}&{}\approx{}&0.080570161309&{}:{}&0.902075806737&{}:{}&0.017354031954&,\\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.106470138742&{}:{}&1.085585763082&{}:{}&0.020884375660&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.432251770720&{}:{}&0.474645386743&{}:{}&0.093102842536&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.090105706697&{}:{}&1.008837226334&{}:{}&-0.098942933031&. \end{alignedat} \]
3b (123)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3b}}&{}\approx{}&-0.117093168886&{}:{}&1.310992966503&{}:{}&-0.193899797618&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&1.000000000000&{}:{}&1.098076211353&{}:{}&-1.098076211353&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.122008467928&{}:{}&1.244016935856&{}:{}&-0.122008467928&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-5.098076211353&{}:{}&5.098076211353&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.238313554719&{}:{}&1.334097441518&{}:{}&-0.095783886799&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.080695238898&{}:{}&1.806952388982&{}:{}&-0.726257150084&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.009807621135&{}:{}&1.229422863406&{}:{}&-0.219615242271&, \end{alignedat} \]
3b (123)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3b}}&{}\approx{}&0.224009237740&{}:{}&1.448018475480&{}:{}&-0.672027713219&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&1.866025403784&{}:{}&-0.866025403784&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.500000000000&{}:{}&0.000000000000&{}:{}&1.500000000000&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.133974596216&{}:{}&0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3b}}&{}\approx{}&-0.138963148813&{}:{}&1.555852595253&{}:{}&-0.416889446440&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.238313554719&{}:{}&1.334097441518&{}:{}&-0.095783886799&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.080695238898&{}:{}&1.806952388982&{}:{}&-0.726257150084&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.009807621135&{}:{}&1.229422863406&{}:{}&-0.219615242271&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&1.866025403784&{}:{}&-0.866025403784&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.500000000000&{}:{}&0.000000000000&{}:{}&1.500000000000&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.133974596216&{}:{}&0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)