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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0a(011)

Malfatti circles

0a (011)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-1.000000000000&{}:{}&1.000000000000&{}:{}&1.000000000000&, \\ P_{\mathbf{a}}&{}\approx{}&-0.477291366137&{}:{}&0.738645683069&{}:{}&0.738645683069&, \\ P^-_{\mathbf{a}}&{}\approx{}&-0.348308999953&{}:{}&0.674154499977&{}:{}&0.674154499977&, \\ P^+_{\mathbf{a}}&{}\approx{}&-0.563652960290&{}:{}&0.781826480145&{}:{}&0.781826480145&, \\ Q_{\mathbf{a}}&{}\approx{}&-0.302169479252&{}:{}&0.651084739626&{}:{}&0.651084739626&, \\ I^\prime_{\mathbf{a}}&{}\approx{}&-0.672027713219&{}:{}&0.836013856610&{}:{}&0.836013856610&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{a}}\)
\(P^-_{\mathbf{a}}\)
\(P^+_{\mathbf{a}}\)
\(Q_{\mathbf{a}}\)
\(I^\prime_{\mathbf{a}}\)
0a (011)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{a}}&{}\approx{}&-2.398717474236&{}:{}&1.699358737118&{}:{}&1.699358737118&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&1.000000000000&{}:{}&0.366025403784&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.366025403784&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.683012701892&{}:{}&0.683012701892&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.726257150084&{}:{}&0.602317462994&{}:{}&1.123939687090&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.726257150084&{}:{}&1.123939687090&{}:{}&0.602317462994&, \\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.224009237740&{}:{}&0.612004618870&{}:{}&0.612004618870&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.608290376865&{}:{}&0.430940107676&{}:{}&1.177350269190&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.608290376865&{}:{}&1.177350269190&{}:{}&0.430940107676&, \\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.500000000000&{}:{}&0.500000000000&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.866025403784&{}:{}&0.000000000000&{}:{}&1.866025403784&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.866025403784&{}:{}&1.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{a}}}{B^\prime_{\mathbf{a}}}{C^\prime_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{a}}}{B^{\prime\prime}_{\mathbf{a}}}{C^{\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{a}}}{B^{\prime\prime\prime}_{\mathbf{a}}}{C^{\prime\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^*_{\mathbf{a}}}{B^*_{\mathbf{a}}}{C^*_{\mathbf{a}}}\)
0a (011)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{a}}}}&{}\approx{}&1.699358737118&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{a}}}}&{}\approx{}&0.366025403784&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{a}}}}&{}\approx{}&0.366025403784&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-1.000000000000&{}:{}&1.000000000000&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-2.398717474236&{}:{}&1.699358737118&{}:{}&1.699358737118&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&1.000000000000&{}:{}&0.366025403784&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.366025403784&{}:{}&1.000000000000&. \end{alignedat} \]
0a (011)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{a}}&{}\approx{}&-0.477291366137&{}:{}&0.738645683069&{}:{}&0.738645683069&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.683012701892&{}:{}&0.683012701892&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.726257150084&{}:{}&0.602317462994&{}:{}&1.123939687090&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.726257150084&{}:{}&1.123939687090&{}:{}&0.602317462994&. \end{alignedat} \]
0a (011)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{a}}&{}\approx{}&-0.348308999953&{}:{}&0.674154499977&{}:{}&0.674154499977&,\\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.224009237740&{}:{}&0.612004618870&{}:{}&0.612004618870&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.608290376865&{}:{}&0.430940107676&{}:{}&1.177350269190&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.608290376865&{}:{}&1.177350269190&{}:{}&0.430940107676&. \end{alignedat} \]
0a (011)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{a}}&{}\approx{}&-0.563652960290&{}:{}&0.781826480145&{}:{}&0.781826480145&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-2.398717474236&{}:{}&1.699358737118&{}:{}&1.699358737118&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&1.000000000000&{}:{}&0.366025403784&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.366025403784&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.683012701892&{}:{}&0.683012701892&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.726257150084&{}:{}&0.602317462994&{}:{}&1.123939687090&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.726257150084&{}:{}&1.123939687090&{}:{}&0.602317462994&, \end{alignedat} \]
0a (011)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{a}}&{}\approx{}&-0.302169479252&{}:{}&0.651084739626&{}:{}&0.651084739626&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.500000000000&{}:{}&0.500000000000&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.866025403784&{}:{}&0.000000000000&{}:{}&1.866025403784&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.866025403784&{}:{}&1.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{a}}&{}\approx{}&-0.672027713219&{}:{}&0.836013856610&{}:{}&0.836013856610&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.683012701892&{}:{}&0.683012701892&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.726257150084&{}:{}&0.602317462994&{}:{}&1.123939687090&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.726257150084&{}:{}&1.123939687090&{}:{}&0.602317462994&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.500000000000&{}:{}&0.500000000000&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.866025403784&{}:{}&0.000000000000&{}:{}&1.866025403784&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.866025403784&{}:{}&1.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)