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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1c (112)

Malfatti circles

1c (112)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.000000000000&{}:{}&1.000000000000&{}:{}&-1.000000000000&, \\ P_{\mathbf{1c}}&{}\approx{}&-0.083834923055&{}:{}&-0.083834923055&{}:{}&1.167669846111&, \\ P^-_{\mathbf{1c}}&{}\approx{}&0.174457630187&{}:{}&0.174457630187&{}:{}&0.651084739626&, \\ P^+_{\mathbf{1c}}&{}\approx{}&-0.577350269190&{}:{}&-0.577350269190&{}:{}&2.154700538379&, \\ Q_{\mathbf{1c}}&{}\approx{}&-6.464101615138&{}:{}&-6.464101615138&{}:{}&13.928203230276&, \\ I^\prime_{\mathbf{1c}}&{}\approx{}&-0.577350269190&{}:{}&-0.577350269190&{}:{}&2.154700538379&, \end{alignedat} \]
\(I_{\mathbf{c}}\) Incenter
\(P_{\mathbf{1c}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{1c}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{1c}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{1c}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{1c}}\) Radical center of the Malfatti circles
1c (112)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1c}}&{}\approx{}&1.000000000000&{}:{}&-1.366025403784&{}:{}&1.366025403784&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.366025403784&{}:{}&1.000000000000&{}:{}&1.366025403784&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-1.366025403784&{}:{}&-1.366025403784&{}:{}&3.732050807569&, \\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.366025403784&{}:{}&-0.183012701892&{}:{}&2.549038105677&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.183012701892&{}:{}&-1.366025403784&{}:{}&2.549038105677&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.183012701892&{}:{}&-0.183012701892&{}:{}&1.366025403784&, \\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.405827419558&{}:{}&0.297086290221&{}:{}&1.108741129337&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.297086290221&{}:{}&-0.405827419558&{}:{}&1.108741129337&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.297086290221&{}:{}&0.297086290221&{}:{}&0.405827419558&, \\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-0.866025403784&{}:{}&1.866025403784&,\\B^*_{\mathbf{1c}}&{}\approx{}&-0.866025403784&{}:{}&0.000000000000&{}:{}&1.866025403784&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1c}}}{B^\prime_{\mathbf{1c}}}{C^\prime_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^*_{\mathbf{1c}}}{B^*_{\mathbf{1c}}}{C^*_{\mathbf{1c}}}\)
1c (112)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1c}}}}&{}\approx{}&-1.366025403784&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1c}}}}&{}\approx{}&-1.366025403784&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1c}}}}&{}\approx{}&-1.366025403784&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.000000000000&{}:{}&1.000000000000&{}:{}&-1.000000000000&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&1.000000000000&{}:{}&-1.366025403784&{}:{}&1.366025403784&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.366025403784&{}:{}&1.000000000000&{}:{}&1.366025403784&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-1.366025403784&{}:{}&-1.366025403784&{}:{}&3.732050807569&. \end{alignedat} \]
1c (112)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1c}}&{}\approx{}&-0.083834923055&{}:{}&-0.083834923055&{}:{}&1.167669846111&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.366025403784&{}:{}&-0.183012701892&{}:{}&2.549038105677&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.183012701892&{}:{}&-1.366025403784&{}:{}&2.549038105677&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.183012701892&{}:{}&-0.183012701892&{}:{}&1.366025403784&. \end{alignedat} \]
1c (112)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1c}}&{}\approx{}&0.174457630187&{}:{}&0.174457630187&{}:{}&0.651084739626&,\\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.405827419558&{}:{}&0.297086290221&{}:{}&1.108741129337&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.297086290221&{}:{}&-0.405827419558&{}:{}&1.108741129337&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.297086290221&{}:{}&0.297086290221&{}:{}&0.405827419558&. \end{alignedat} \]
1c (112)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1c}}&{}\approx{}&-0.577350269190&{}:{}&-0.577350269190&{}:{}&2.154700538379&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&1.000000000000&{}:{}&-1.366025403784&{}:{}&1.366025403784&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.366025403784&{}:{}&1.000000000000&{}:{}&1.366025403784&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-1.366025403784&{}:{}&-1.366025403784&{}:{}&3.732050807569&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.366025403784&{}:{}&-0.183012701892&{}:{}&2.549038105677&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.183012701892&{}:{}&-1.366025403784&{}:{}&2.549038105677&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.183012701892&{}:{}&-0.183012701892&{}:{}&1.366025403784&, \end{alignedat} \]
1c (112)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1c}}&{}\approx{}&-6.464101615138&{}:{}&-6.464101615138&{}:{}&13.928203230276&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-0.866025403784&{}:{}&1.866025403784&,\\B^*_{\mathbf{1c}}&{}\approx{}&-0.866025403784&{}:{}&0.000000000000&{}:{}&1.866025403784&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1c}}&{}\approx{}&-0.577350269190&{}:{}&-0.577350269190&{}:{}&2.154700538379&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.366025403784&{}:{}&-0.183012701892&{}:{}&2.549038105677&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.183012701892&{}:{}&-1.366025403784&{}:{}&2.549038105677&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.183012701892&{}:{}&-0.183012701892&{}:{}&1.366025403784&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-0.866025403784&{}:{}&1.866025403784&,\\B^*_{\mathbf{1c}}&{}\approx{}&-0.866025403784&{}:{}&0.000000000000&{}:{}&1.866025403784&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)