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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3c(132)

Malfatti circles

3c (132)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.000000000000&{}:{}&1.000000000000&{}:{}&-1.000000000000&, \\ P_{\mathbf{3c}}&{}\approx{}&-0.008668979678&{}:{}&-0.078020817101&{}:{}&1.086689796779&, \\ P^-_{\mathbf{3c}}&{}\approx{}&0.080570161309&{}:{}&0.017354031954&{}:{}&0.902075806737&, \\ P^+_{\mathbf{3c}}&{}\approx{}&-0.117093168886&{}:{}&-0.193899797618&{}:{}&1.310992966503&, \\ Q_{\mathbf{3c}}&{}\approx{}&0.224009237740&{}:{}&-0.672027713219&{}:{}&1.448018475480&, \\ I^\prime_{\mathbf{3c}}&{}\approx{}&-0.138963148813&{}:{}&-0.416889446440&{}:{}&1.555852595253&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{3c}}\)
\(P^-_{\mathbf{3c}}\)
\(P^+_{\mathbf{3c}}\)
\(Q_{\mathbf{3c}}\)
\(I^\prime_{\mathbf{3c}}\)
3c (132)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3c}}&{}\approx{}&1.000000000000&{}:{}&-1.098076211353&{}:{}&1.098076211353&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-5.098076211353&{}:{}&1.000000000000&{}:{}&5.098076211353&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.122008467928&{}:{}&-0.122008467928&{}:{}&1.244016935856&, \\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.238313554719&{}:{}&-0.095783886799&{}:{}&1.334097441518&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.009807621135&{}:{}&-0.219615242271&{}:{}&1.229422863406&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.080695238898&{}:{}&-0.726257150084&{}:{}&1.806952388982&, \\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.106470138742&{}:{}&0.020884375660&{}:{}&1.085585763082&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.090105706697&{}:{}&-0.098942933031&{}:{}&1.008837226334&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.432251770720&{}:{}&0.093102842536&{}:{}&0.474645386743&, \\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-0.866025403784&{}:{}&1.866025403784&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.133974596216&{}:{}&0.000000000000&{}:{}&0.866025403784&,\\C^*_{\mathbf{3c}}&{}\approx{}&-0.500000000000&{}:{}&1.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3c}}}{B^\prime_{\mathbf{3c}}}{C^\prime_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^*_{\mathbf{3c}}}{B^*_{\mathbf{3c}}}{C^*_{\mathbf{3c}}}\)
3c (132)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3c}}}}&{}\approx{}&-1.098076211353&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3c}}}}&{}\approx{}&-5.098076211353&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.122008467928&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.000000000000&{}:{}&1.000000000000&{}:{}&-1.000000000000&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&1.000000000000&{}:{}&-1.098076211353&{}:{}&1.098076211353&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-5.098076211353&{}:{}&1.000000000000&{}:{}&5.098076211353&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.122008467928&{}:{}&-0.122008467928&{}:{}&1.244016935856&. \end{alignedat} \]
3c (132)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3c}}&{}\approx{}&-0.008668979678&{}:{}&-0.078020817101&{}:{}&1.086689796779&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.238313554719&{}:{}&-0.095783886799&{}:{}&1.334097441518&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.009807621135&{}:{}&-0.219615242271&{}:{}&1.229422863406&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.080695238898&{}:{}&-0.726257150084&{}:{}&1.806952388982&. \end{alignedat} \]
3c (132)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3c}}&{}\approx{}&0.080570161309&{}:{}&0.017354031954&{}:{}&0.902075806737&,\\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.106470138742&{}:{}&0.020884375660&{}:{}&1.085585763082&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.090105706697&{}:{}&-0.098942933031&{}:{}&1.008837226334&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.432251770720&{}:{}&0.093102842536&{}:{}&0.474645386743&. \end{alignedat} \]
3c (132)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3c}}&{}\approx{}&-0.117093168886&{}:{}&-0.193899797618&{}:{}&1.310992966503&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&1.000000000000&{}:{}&-1.098076211353&{}:{}&1.098076211353&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-5.098076211353&{}:{}&1.000000000000&{}:{}&5.098076211353&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.122008467928&{}:{}&-0.122008467928&{}:{}&1.244016935856&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.238313554719&{}:{}&-0.095783886799&{}:{}&1.334097441518&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.009807621135&{}:{}&-0.219615242271&{}:{}&1.229422863406&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.080695238898&{}:{}&-0.726257150084&{}:{}&1.806952388982&, \end{alignedat} \]
3c (132)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3c}}&{}\approx{}&0.224009237740&{}:{}&-0.672027713219&{}:{}&1.448018475480&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-0.866025403784&{}:{}&1.866025403784&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.133974596216&{}:{}&0.000000000000&{}:{}&0.866025403784&,\\C^*_{\mathbf{3c}}&{}\approx{}&-0.500000000000&{}:{}&1.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3c}}&{}\approx{}&-0.138963148813&{}:{}&-0.416889446440&{}:{}&1.555852595253&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.238313554719&{}:{}&-0.095783886799&{}:{}&1.334097441518&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.009807621135&{}:{}&-0.219615242271&{}:{}&1.229422863406&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.080695238898&{}:{}&-0.726257150084&{}:{}&1.806952388982&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-0.866025403784&{}:{}&1.866025403784&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.133974596216&{}:{}&0.000000000000&{}:{}&0.866025403784&,\\C^*_{\mathbf{3c}}&{}\approx{}&-0.500000000000&{}:{}&1.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)