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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6c(330)

Malfatti circles

6c (330)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.000000000000&{}:{}&1.000000000000&{}:{}&-1.000000000000&, \\ P_{\mathbf{6c}}&{}\approx{}&0.518617531751&{}:{}&0.518617531751&{}:{}&-0.037235063502&, \\ P^-_{\mathbf{6c}}&{}\approx{}&0.440926985198&{}:{}&0.440926985198&{}:{}&0.118146029605&, \\ P^+_{\mathbf{6c}}&{}\approx{}&0.577350269190&{}:{}&0.577350269190&{}:{}&-0.154700538379&, \\ Q_{\mathbf{6c}}&{}\approx{}&0.464101615138&{}:{}&0.464101615138&{}:{}&0.071796769724&, \\ I^\prime_{\mathbf{6c}}&{}\approx{}&0.577350269190&{}:{}&0.577350269190&{}:{}&-0.154700538379&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{6c}}\)
\(P^-_{\mathbf{6c}}\)
\(P^+_{\mathbf{6c}}\)
\(Q_{\mathbf{6c}}\)
\(I^\prime_{\mathbf{6c}}\)
6c (330)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6c}}&{}\approx{}&1.000000000000&{}:{}&0.366025403784&{}:{}&-0.366025403784&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.366025403784&{}:{}&1.000000000000&{}:{}&-0.366025403784&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.366025403784&{}:{}&0.366025403784&{}:{}&0.267949192431&, \\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.366025403784&{}:{}&0.683012701892&{}:{}&-0.049038105677&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.683012701892&{}:{}&0.366025403784&{}:{}&-0.049038105677&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.683012701892&{}:{}&0.683012701892&{}:{}&-0.366025403784&, \\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.224009237740&{}:{}&0.612004618870&{}:{}&0.163986143390&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.612004618870&{}:{}&0.224009237740&{}:{}&0.163986143390&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.612004618870&{}:{}&0.612004618870&{}:{}&-0.224009237740&, \\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.866025403784&{}:{}&0.133974596216&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.866025403784&{}:{}&0.000000000000&{}:{}&0.133974596216&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6c}}}{B^\prime_{\mathbf{6c}}}{C^\prime_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^*_{\mathbf{6c}}}{B^*_{\mathbf{6c}}}{C^*_{\mathbf{6c}}}\)
6c (330)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6c}}}}&{}\approx{}&0.366025403784&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6c}}}}&{}\approx{}&0.366025403784&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6c}}}}&{}\approx{}&0.366025403784&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.000000000000&{}:{}&1.000000000000&{}:{}&-1.000000000000&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.000000000000&{}:{}&0.366025403784&{}:{}&-0.366025403784&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.366025403784&{}:{}&1.000000000000&{}:{}&-0.366025403784&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.366025403784&{}:{}&0.366025403784&{}:{}&0.267949192431&. \end{alignedat} \]
6c (330)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6c}}&{}\approx{}&0.518617531751&{}:{}&0.518617531751&{}:{}&-0.037235063502&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.366025403784&{}:{}&0.683012701892&{}:{}&-0.049038105677&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.683012701892&{}:{}&0.366025403784&{}:{}&-0.049038105677&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.683012701892&{}:{}&0.683012701892&{}:{}&-0.366025403784&. \end{alignedat} \]
6c (330)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6c}}&{}\approx{}&0.440926985198&{}:{}&0.440926985198&{}:{}&0.118146029605&,\\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.224009237740&{}:{}&0.612004618870&{}:{}&0.163986143390&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.612004618870&{}:{}&0.224009237740&{}:{}&0.163986143390&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.612004618870&{}:{}&0.612004618870&{}:{}&-0.224009237740&. \end{alignedat} \]
6c (330)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6c}}&{}\approx{}&0.577350269190&{}:{}&0.577350269190&{}:{}&-0.154700538379&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.000000000000&{}:{}&0.366025403784&{}:{}&-0.366025403784&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.366025403784&{}:{}&1.000000000000&{}:{}&-0.366025403784&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.366025403784&{}:{}&0.366025403784&{}:{}&0.267949192431&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.366025403784&{}:{}&0.683012701892&{}:{}&-0.049038105677&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.683012701892&{}:{}&0.366025403784&{}:{}&-0.049038105677&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.683012701892&{}:{}&0.683012701892&{}:{}&-0.366025403784&, \end{alignedat} \]
6c (330)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6c}}&{}\approx{}&0.464101615138&{}:{}&0.464101615138&{}:{}&0.071796769724&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.866025403784&{}:{}&0.133974596216&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.866025403784&{}:{}&0.000000000000&{}:{}&0.133974596216&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6c}}&{}\approx{}&0.577350269190&{}:{}&0.577350269190&{}:{}&-0.154700538379&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.366025403784&{}:{}&0.683012701892&{}:{}&-0.049038105677&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.683012701892&{}:{}&0.366025403784&{}:{}&-0.049038105677&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.683012701892&{}:{}&0.683012701892&{}:{}&-0.366025403784&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.866025403784&{}:{}&0.133974596216&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.866025403784&{}:{}&0.000000000000&{}:{}&0.133974596216&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)