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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4a(211)

Malfatti circles

4a (211)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-1.000000000000&{}:{}&1.000000000000&{}:{}&1.000000000000&, \\ P_{\mathbf{4a}}&{}\approx{}&1.167669846111&{}:{}&-0.083834923055&{}:{}&-0.083834923055&, \\ P^-_{\mathbf{4a}}&{}\approx{}&0.651084739626&{}:{}&0.174457630187&{}:{}&0.174457630187&, \\ P^+_{\mathbf{4a}}&{}\approx{}&2.154700538379&{}:{}&-0.577350269190&{}:{}&-0.577350269190&, \\ Q_{\mathbf{4a}}&{}\approx{}&13.928203230276&{}:{}&-6.464101615138&{}:{}&-6.464101615138&, \\ I^\prime_{\mathbf{4a}}&{}\approx{}&2.154700538379&{}:{}&-0.577350269190&{}:{}&-0.577350269190&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{4a}}\)
\(P^-_{\mathbf{4a}}\)
\(P^+_{\mathbf{4a}}\)
\(Q_{\mathbf{4a}}\)
\(I^\prime_{\mathbf{4a}}\)
4a (211)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4a}}&{}\approx{}&3.732050807569&{}:{}&-1.366025403784&{}:{}&-1.366025403784&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&1.366025403784&{}:{}&1.000000000000&{}:{}&-1.366025403784&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&1.366025403784&{}:{}&-1.366025403784&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.366025403784&{}:{}&-0.183012701892&{}:{}&-0.183012701892&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.549038105677&{}:{}&-1.366025403784&{}:{}&-0.183012701892&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.549038105677&{}:{}&-0.183012701892&{}:{}&-1.366025403784&, \\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.405827419558&{}:{}&0.297086290221&{}:{}&0.297086290221&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.108741129337&{}:{}&-0.405827419558&{}:{}&0.297086290221&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.108741129337&{}:{}&0.297086290221&{}:{}&-0.405827419558&, \\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.500000000000&{}:{}&0.500000000000&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.866025403784&{}:{}&0.000000000000&{}:{}&-0.866025403784&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.866025403784&{}:{}&-0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4a}}}{B^\prime_{\mathbf{4a}}}{C^\prime_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^*_{\mathbf{4a}}}{B^*_{\mathbf{4a}}}{C^*_{\mathbf{4a}}}\)
4a (211)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4a}}}}&{}\approx{}&-1.366025403784&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4a}}}}&{}\approx{}&-1.366025403784&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4a}}}}&{}\approx{}&-1.366025403784&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-1.000000000000&{}:{}&1.000000000000&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&3.732050807569&{}:{}&-1.366025403784&{}:{}&-1.366025403784&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&1.366025403784&{}:{}&1.000000000000&{}:{}&-1.366025403784&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&1.366025403784&{}:{}&-1.366025403784&{}:{}&1.000000000000&. \end{alignedat} \]
4a (211)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4a}}&{}\approx{}&1.167669846111&{}:{}&-0.083834923055&{}:{}&-0.083834923055&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.366025403784&{}:{}&-0.183012701892&{}:{}&-0.183012701892&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.549038105677&{}:{}&-1.366025403784&{}:{}&-0.183012701892&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.549038105677&{}:{}&-0.183012701892&{}:{}&-1.366025403784&. \end{alignedat} \]
4a (211)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4a}}&{}\approx{}&0.651084739626&{}:{}&0.174457630187&{}:{}&0.174457630187&,\\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.405827419558&{}:{}&0.297086290221&{}:{}&0.297086290221&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.108741129337&{}:{}&-0.405827419558&{}:{}&0.297086290221&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.108741129337&{}:{}&0.297086290221&{}:{}&-0.405827419558&. \end{alignedat} \]
4a (211)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4a}}&{}\approx{}&2.154700538379&{}:{}&-0.577350269190&{}:{}&-0.577350269190&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&3.732050807569&{}:{}&-1.366025403784&{}:{}&-1.366025403784&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&1.366025403784&{}:{}&1.000000000000&{}:{}&-1.366025403784&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&1.366025403784&{}:{}&-1.366025403784&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.366025403784&{}:{}&-0.183012701892&{}:{}&-0.183012701892&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.549038105677&{}:{}&-1.366025403784&{}:{}&-0.183012701892&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.549038105677&{}:{}&-0.183012701892&{}:{}&-1.366025403784&, \end{alignedat} \]
4a (211)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4a}}&{}\approx{}&13.928203230276&{}:{}&-6.464101615138&{}:{}&-6.464101615138&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.500000000000&{}:{}&0.500000000000&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.866025403784&{}:{}&0.000000000000&{}:{}&-0.866025403784&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.866025403784&{}:{}&-0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4a}}&{}\approx{}&2.154700538379&{}:{}&-0.577350269190&{}:{}&-0.577350269190&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.366025403784&{}:{}&-0.183012701892&{}:{}&-0.183012701892&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.549038105677&{}:{}&-1.366025403784&{}:{}&-0.183012701892&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.549038105677&{}:{}&-0.183012701892&{}:{}&-1.366025403784&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.500000000000&{}:{}&0.500000000000&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.866025403784&{}:{}&0.000000000000&{}:{}&-0.866025403784&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.866025403784&{}:{}&-0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)