Derousseau's Generalization of the Malfatti circles

Equilateral Triangle

\(a:b:c=1:1:1\), \(A=60\degree\), \(B=60\degree\), \(C=60\degree\).


[Top] > Equilateral Triangle > 6b (321)

6b(321)

Malfatti circles

6b (321)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&1.000000000000&{}:{}&-1.000000000000&{}:{}&1.000000000000&, \\ P_{\mathbf{6b}}&{}\approx{}&-0.078020817101&{}:{}&1.086689796779&{}:{}&-0.008668979678&, \\ P^-_{\mathbf{6b}}&{}\approx{}&0.017354031954&{}:{}&0.902075806737&{}:{}&0.080570161309&, \\ P^+_{\mathbf{6b}}&{}\approx{}&-0.193899797618&{}:{}&1.310992966503&{}:{}&-0.117093168886&, \\ Q_{\mathbf{6b}}&{}\approx{}&-0.672027713219&{}:{}&1.448018475480&{}:{}&0.224009237740&, \\ I^\prime_{\mathbf{6b}}&{}\approx{}&-0.416889446440&{}:{}&1.555852595253&{}:{}&-0.138963148813&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{6b}}\)
\(P^-_{\mathbf{6b}}\)
\(P^+_{\mathbf{6b}}\)
\(Q_{\mathbf{6b}}\)
\(I^\prime_{\mathbf{6b}}\)
6b (321)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6b}}&{}\approx{}&1.000000000000&{}:{}&5.098076211353&{}:{}&-5.098076211353&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.122008467928&{}:{}&1.244016935856&{}:{}&-0.122008467928&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-1.098076211353&{}:{}&1.098076211353&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.219615242271&{}:{}&1.229422863406&{}:{}&-0.009807621135&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.726257150084&{}:{}&1.806952388982&{}:{}&-0.080695238898&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.095783886799&{}:{}&1.334097441518&{}:{}&-0.238313554719&, \\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.098942933031&{}:{}&1.008837226334&{}:{}&0.090105706697&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.093102842536&{}:{}&0.474645386743&{}:{}&0.432251770720&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.020884375660&{}:{}&1.085585763082&{}:{}&-0.106470138742&, \\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.866025403784&{}:{}&0.133974596216&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.500000000000&{}:{}&0.000000000000&{}:{}&-0.500000000000&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.866025403784&{}:{}&1.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6b}}}{B^\prime_{\mathbf{6b}}}{C^\prime_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^*_{\mathbf{6b}}}{B^*_{\mathbf{6b}}}{C^*_{\mathbf{6b}}}\)
6b (321)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6b}}}}&{}\approx{}&-5.098076211353&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6b}}}}&{}\approx{}&-0.122008467928&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6b}}}}&{}\approx{}&-1.098076211353&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&1.000000000000&{}:{}&-1.000000000000&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&1.000000000000&{}:{}&5.098076211353&{}:{}&-5.098076211353&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.122008467928&{}:{}&1.244016935856&{}:{}&-0.122008467928&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-1.098076211353&{}:{}&1.098076211353&{}:{}&1.000000000000&. \end{alignedat} \]
6b (321)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6b}}&{}\approx{}&-0.078020817101&{}:{}&1.086689796779&{}:{}&-0.008668979678&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.219615242271&{}:{}&1.229422863406&{}:{}&-0.009807621135&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.726257150084&{}:{}&1.806952388982&{}:{}&-0.080695238898&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.095783886799&{}:{}&1.334097441518&{}:{}&-0.238313554719&. \end{alignedat} \]
6b (321)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6b}}&{}\approx{}&0.017354031954&{}:{}&0.902075806737&{}:{}&0.080570161309&,\\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.098942933031&{}:{}&1.008837226334&{}:{}&0.090105706697&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.093102842536&{}:{}&0.474645386743&{}:{}&0.432251770720&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.020884375660&{}:{}&1.085585763082&{}:{}&-0.106470138742&. \end{alignedat} \]
6b (321)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6b}}&{}\approx{}&-0.193899797618&{}:{}&1.310992966503&{}:{}&-0.117093168886&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&1.000000000000&{}:{}&5.098076211353&{}:{}&-5.098076211353&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.122008467928&{}:{}&1.244016935856&{}:{}&-0.122008467928&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-1.098076211353&{}:{}&1.098076211353&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.219615242271&{}:{}&1.229422863406&{}:{}&-0.009807621135&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.726257150084&{}:{}&1.806952388982&{}:{}&-0.080695238898&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.095783886799&{}:{}&1.334097441518&{}:{}&-0.238313554719&, \end{alignedat} \]
6b (321)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6b}}&{}\approx{}&-0.672027713219&{}:{}&1.448018475480&{}:{}&0.224009237740&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.866025403784&{}:{}&0.133974596216&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.500000000000&{}:{}&0.000000000000&{}:{}&-0.500000000000&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.866025403784&{}:{}&1.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6b}}&{}\approx{}&-0.416889446440&{}:{}&1.555852595253&{}:{}&-0.138963148813&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.219615242271&{}:{}&1.229422863406&{}:{}&-0.009807621135&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.726257150084&{}:{}&1.806952388982&{}:{}&-0.080695238898&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.095783886799&{}:{}&1.334097441518&{}:{}&-0.238313554719&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.866025403784&{}:{}&0.133974596216&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.500000000000&{}:{}&0.000000000000&{}:{}&-0.500000000000&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.866025403784&{}:{}&1.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)