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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0c(110)

Malfatti circles

0c (110)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.000000000000&{}:{}&1.000000000000&{}:{}&-1.000000000000&, \\ P_{\mathbf{c}}&{}\approx{}&0.738645683069&{}:{}&0.738645683069&{}:{}&-0.477291366137&, \\ P^-_{\mathbf{c}}&{}\approx{}&0.674154499977&{}:{}&0.674154499977&{}:{}&-0.348308999953&, \\ P^+_{\mathbf{c}}&{}\approx{}&0.781826480145&{}:{}&0.781826480145&{}:{}&-0.563652960290&, \\ Q_{\mathbf{c}}&{}\approx{}&0.651084739626&{}:{}&0.651084739626&{}:{}&-0.302169479252&, \\ I^\prime_{\mathbf{c}}&{}\approx{}&0.836013856610&{}:{}&0.836013856610&{}:{}&-0.672027713219&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{c}}\)
\(P^-_{\mathbf{c}}\)
\(P^+_{\mathbf{c}}\)
\(Q_{\mathbf{c}}\)
\(I^\prime_{\mathbf{c}}\)
0c (110)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{c}}&{}\approx{}&1.000000000000&{}:{}&0.366025403784&{}:{}&-0.366025403784&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.366025403784&{}:{}&1.000000000000&{}:{}&-0.366025403784&,\\C^\prime_{\mathbf{c}}&{}\approx{}&1.699358737118&{}:{}&1.699358737118&{}:{}&-2.398717474236&, \\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.602317462994&{}:{}&1.123939687090&{}:{}&-0.726257150084&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.123939687090&{}:{}&0.602317462994&{}:{}&-0.726257150084&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.683012701892&{}:{}&0.683012701892&{}:{}&-0.366025403784&, \\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.430940107676&{}:{}&1.177350269190&{}:{}&-0.608290376865&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.177350269190&{}:{}&0.430940107676&{}:{}&-0.608290376865&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.612004618870&{}:{}&0.612004618870&{}:{}&-0.224009237740&, \\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&1.866025403784&{}:{}&-0.866025403784&,\\B^*_{\mathbf{c}}&{}\approx{}&1.866025403784&{}:{}&0.000000000000&{}:{}&-0.866025403784&,\\C^*_{\mathbf{c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{c}}}{B^\prime_{\mathbf{c}}}{C^\prime_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{c}}}{B^{\prime\prime}_{\mathbf{c}}}{C^{\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{c}}}{B^{\prime\prime\prime}_{\mathbf{c}}}{C^{\prime\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^*_{\mathbf{c}}}{B^*_{\mathbf{c}}}{C^*_{\mathbf{c}}}\)
0c (110)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{c}}}}&{}\approx{}&0.366025403784&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{c}}}}&{}\approx{}&0.366025403784&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{c}}}}&{}\approx{}&1.699358737118&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.000000000000&{}:{}&1.000000000000&{}:{}&-1.000000000000&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.000000000000&{}:{}&0.366025403784&{}:{}&-0.366025403784&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.366025403784&{}:{}&1.000000000000&{}:{}&-0.366025403784&,\\C^\prime_{\mathbf{c}}&{}\approx{}&1.699358737118&{}:{}&1.699358737118&{}:{}&-2.398717474236&. \end{alignedat} \]
0c (110)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{c}}&{}\approx{}&0.738645683069&{}:{}&0.738645683069&{}:{}&-0.477291366137&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.602317462994&{}:{}&1.123939687090&{}:{}&-0.726257150084&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.123939687090&{}:{}&0.602317462994&{}:{}&-0.726257150084&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.683012701892&{}:{}&0.683012701892&{}:{}&-0.366025403784&. \end{alignedat} \]
0c (110)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{c}}&{}\approx{}&0.674154499977&{}:{}&0.674154499977&{}:{}&-0.348308999953&,\\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.430940107676&{}:{}&1.177350269190&{}:{}&-0.608290376865&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.177350269190&{}:{}&0.430940107676&{}:{}&-0.608290376865&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.612004618870&{}:{}&0.612004618870&{}:{}&-0.224009237740&. \end{alignedat} \]
0c (110)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{c}}&{}\approx{}&0.781826480145&{}:{}&0.781826480145&{}:{}&-0.563652960290&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.000000000000&{}:{}&0.366025403784&{}:{}&-0.366025403784&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.366025403784&{}:{}&1.000000000000&{}:{}&-0.366025403784&,\\C^\prime_{\mathbf{c}}&{}\approx{}&1.699358737118&{}:{}&1.699358737118&{}:{}&-2.398717474236&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.602317462994&{}:{}&1.123939687090&{}:{}&-0.726257150084&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.123939687090&{}:{}&0.602317462994&{}:{}&-0.726257150084&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.683012701892&{}:{}&0.683012701892&{}:{}&-0.366025403784&, \end{alignedat} \]
0c (110)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{c}}&{}\approx{}&0.651084739626&{}:{}&0.651084739626&{}:{}&-0.302169479252&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&1.866025403784&{}:{}&-0.866025403784&,\\B^*_{\mathbf{c}}&{}\approx{}&1.866025403784&{}:{}&0.000000000000&{}:{}&-0.866025403784&,\\C^*_{\mathbf{c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{c}}&{}\approx{}&0.836013856610&{}:{}&0.836013856610&{}:{}&-0.672027713219&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.602317462994&{}:{}&1.123939687090&{}:{}&-0.726257150084&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.123939687090&{}:{}&0.602317462994&{}:{}&-0.726257150084&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.683012701892&{}:{}&0.683012701892&{}:{}&-0.366025403784&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&1.866025403784&{}:{}&-0.866025403784&,\\B^*_{\mathbf{c}}&{}\approx{}&1.866025403784&{}:{}&0.000000000000&{}:{}&-0.866025403784&,\\C^*_{\mathbf{c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)