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 > 0b (101)

0b(101)

Malfatti circles

0b (101)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&1.000000000000&{}:{}&-1.000000000000&{}:{}&1.000000000000&, \\ P_{\mathbf{b}}&{}\approx{}&0.738645683069&{}:{}&-0.477291366137&{}:{}&0.738645683069&, \\ P^-_{\mathbf{b}}&{}\approx{}&0.674154499977&{}:{}&-0.348308999953&{}:{}&0.674154499977&, \\ P^+_{\mathbf{b}}&{}\approx{}&0.781826480145&{}:{}&-0.563652960290&{}:{}&0.781826480145&, \\ Q_{\mathbf{b}}&{}\approx{}&0.651084739626&{}:{}&-0.302169479252&{}:{}&0.651084739626&, \\ I^\prime_{\mathbf{b}}&{}\approx{}&0.836013856610&{}:{}&-0.672027713219&{}:{}&0.836013856610&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{b}}\)
\(P^-_{\mathbf{b}}\)
\(P^+_{\mathbf{b}}\)
\(Q_{\mathbf{b}}\)
\(I^\prime_{\mathbf{b}}\)
0b (101)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{b}}&{}\approx{}&1.000000000000&{}:{}&-0.366025403784&{}:{}&0.366025403784&,\\B^\prime_{\mathbf{b}}&{}\approx{}&1.699358737118&{}:{}&-2.398717474236&{}:{}&1.699358737118&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.366025403784&{}:{}&-0.366025403784&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.602317462994&{}:{}&-0.726257150084&{}:{}&1.123939687090&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.683012701892&{}:{}&-0.366025403784&{}:{}&0.683012701892&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&1.123939687090&{}:{}&-0.726257150084&{}:{}&0.602317462994&, \\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.430940107676&{}:{}&-0.608290376865&{}:{}&1.177350269190&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.612004618870&{}:{}&-0.224009237740&{}:{}&0.612004618870&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&1.177350269190&{}:{}&-0.608290376865&{}:{}&0.430940107676&, \\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-0.866025403784&{}:{}&1.866025403784&,\\B^*_{\mathbf{b}}&{}\approx{}&0.500000000000&{}:{}&0.000000000000&{}:{}&0.500000000000&,\\C^*_{\mathbf{b}}&{}\approx{}&1.866025403784&{}:{}&-0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{b}}}{B^\prime_{\mathbf{b}}}{C^\prime_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{b}}}{B^{\prime\prime}_{\mathbf{b}}}{C^{\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{b}}}{B^{\prime\prime\prime}_{\mathbf{b}}}{C^{\prime\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^*_{\mathbf{b}}}{B^*_{\mathbf{b}}}{C^*_{\mathbf{b}}}\)
0b (101)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{b}}}}&{}\approx{}&0.366025403784&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{b}}}}&{}\approx{}&1.699358737118&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{b}}}}&{}\approx{}&0.366025403784&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&1.000000000000&{}:{}&-1.000000000000&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&1.000000000000&{}:{}&-0.366025403784&{}:{}&0.366025403784&,\\B^\prime_{\mathbf{b}}&{}\approx{}&1.699358737118&{}:{}&-2.398717474236&{}:{}&1.699358737118&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.366025403784&{}:{}&-0.366025403784&{}:{}&1.000000000000&. \end{alignedat} \]
0b (101)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{b}}&{}\approx{}&0.738645683069&{}:{}&-0.477291366137&{}:{}&0.738645683069&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.602317462994&{}:{}&-0.726257150084&{}:{}&1.123939687090&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.683012701892&{}:{}&-0.366025403784&{}:{}&0.683012701892&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&1.123939687090&{}:{}&-0.726257150084&{}:{}&0.602317462994&. \end{alignedat} \]
0b (101)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{b}}&{}\approx{}&0.674154499977&{}:{}&-0.348308999953&{}:{}&0.674154499977&,\\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.430940107676&{}:{}&-0.608290376865&{}:{}&1.177350269190&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.612004618870&{}:{}&-0.224009237740&{}:{}&0.612004618870&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&1.177350269190&{}:{}&-0.608290376865&{}:{}&0.430940107676&. \end{alignedat} \]
0b (101)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{b}}&{}\approx{}&0.781826480145&{}:{}&-0.563652960290&{}:{}&0.781826480145&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&1.000000000000&{}:{}&-0.366025403784&{}:{}&0.366025403784&,\\B^\prime_{\mathbf{b}}&{}\approx{}&1.699358737118&{}:{}&-2.398717474236&{}:{}&1.699358737118&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.366025403784&{}:{}&-0.366025403784&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.602317462994&{}:{}&-0.726257150084&{}:{}&1.123939687090&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.683012701892&{}:{}&-0.366025403784&{}:{}&0.683012701892&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&1.123939687090&{}:{}&-0.726257150084&{}:{}&0.602317462994&, \end{alignedat} \]
0b (101)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{b}}&{}\approx{}&0.651084739626&{}:{}&-0.302169479252&{}:{}&0.651084739626&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-0.866025403784&{}:{}&1.866025403784&,\\B^*_{\mathbf{b}}&{}\approx{}&0.500000000000&{}:{}&0.000000000000&{}:{}&0.500000000000&,\\C^*_{\mathbf{b}}&{}\approx{}&1.866025403784&{}:{}&-0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{b}}&{}\approx{}&0.836013856610&{}:{}&-0.672027713219&{}:{}&0.836013856610&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.602317462994&{}:{}&-0.726257150084&{}:{}&1.123939687090&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.683012701892&{}:{}&-0.366025403784&{}:{}&0.683012701892&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&1.123939687090&{}:{}&-0.726257150084&{}:{}&0.602317462994&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-0.866025403784&{}:{}&1.866025403784&,\\B^*_{\mathbf{b}}&{}\approx{}&0.500000000000&{}:{}&0.000000000000&{}:{}&0.500000000000&,\\C^*_{\mathbf{b}}&{}\approx{}&1.866025403784&{}:{}&-0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)