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 > 3a (033)

3a(033)

Malfatti circles

3a (033)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-1.000000000000&{}:{}&1.000000000000&{}:{}&1.000000000000&, \\ P_{\mathbf{3a}}&{}\approx{}&-0.037235063502&{}:{}&0.518617531751&{}:{}&0.518617531751&, \\ P^-_{\mathbf{3a}}&{}\approx{}&0.118146029605&{}:{}&0.440926985198&{}:{}&0.440926985198&, \\ P^+_{\mathbf{3a}}&{}\approx{}&-0.154700538379&{}:{}&0.577350269190&{}:{}&0.577350269190&, \\ Q_{\mathbf{3a}}&{}\approx{}&0.071796769724&{}:{}&0.464101615138&{}:{}&0.464101615138&, \\ I^\prime_{\mathbf{3a}}&{}\approx{}&-0.154700538379&{}:{}&0.577350269190&{}:{}&0.577350269190&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{3a}}\)
\(P^-_{\mathbf{3a}}\)
\(P^+_{\mathbf{3a}}\)
\(Q_{\mathbf{3a}}\)
\(I^\prime_{\mathbf{3a}}\)
3a (033)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3a}}&{}\approx{}&0.267949192431&{}:{}&0.366025403784&{}:{}&0.366025403784&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.366025403784&{}:{}&1.000000000000&{}:{}&0.366025403784&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.366025403784&{}:{}&0.366025403784&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.366025403784&{}:{}&0.683012701892&{}:{}&0.683012701892&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.049038105677&{}:{}&0.366025403784&{}:{}&0.683012701892&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.049038105677&{}:{}&0.683012701892&{}:{}&0.366025403784&, \\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.224009237740&{}:{}&0.612004618870&{}:{}&0.612004618870&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.163986143390&{}:{}&0.224009237740&{}:{}&0.612004618870&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.163986143390&{}:{}&0.612004618870&{}:{}&0.224009237740&, \\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.500000000000&{}:{}&0.500000000000&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.133974596216&{}:{}&0.000000000000&{}:{}&0.866025403784&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.133974596216&{}:{}&0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3a}}}{B^\prime_{\mathbf{3a}}}{C^\prime_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^*_{\mathbf{3a}}}{B^*_{\mathbf{3a}}}{C^*_{\mathbf{3a}}}\)
3a (033)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3a}}}}&{}\approx{}&0.366025403784&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3a}}}}&{}\approx{}&0.366025403784&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3a}}}}&{}\approx{}&0.366025403784&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-1.000000000000&{}:{}&1.000000000000&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.267949192431&{}:{}&0.366025403784&{}:{}&0.366025403784&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.366025403784&{}:{}&1.000000000000&{}:{}&0.366025403784&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.366025403784&{}:{}&0.366025403784&{}:{}&1.000000000000&. \end{alignedat} \]
3a (033)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3a}}&{}\approx{}&-0.037235063502&{}:{}&0.518617531751&{}:{}&0.518617531751&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.366025403784&{}:{}&0.683012701892&{}:{}&0.683012701892&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.049038105677&{}:{}&0.366025403784&{}:{}&0.683012701892&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.049038105677&{}:{}&0.683012701892&{}:{}&0.366025403784&. \end{alignedat} \]
3a (033)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3a}}&{}\approx{}&0.118146029605&{}:{}&0.440926985198&{}:{}&0.440926985198&,\\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.224009237740&{}:{}&0.612004618870&{}:{}&0.612004618870&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.163986143390&{}:{}&0.224009237740&{}:{}&0.612004618870&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.163986143390&{}:{}&0.612004618870&{}:{}&0.224009237740&. \end{alignedat} \]
3a (033)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3a}}&{}\approx{}&-0.154700538379&{}:{}&0.577350269190&{}:{}&0.577350269190&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.267949192431&{}:{}&0.366025403784&{}:{}&0.366025403784&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.366025403784&{}:{}&1.000000000000&{}:{}&0.366025403784&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.366025403784&{}:{}&0.366025403784&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.366025403784&{}:{}&0.683012701892&{}:{}&0.683012701892&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.049038105677&{}:{}&0.366025403784&{}:{}&0.683012701892&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.049038105677&{}:{}&0.683012701892&{}:{}&0.366025403784&, \end{alignedat} \]
3a (033)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3a}}&{}\approx{}&0.071796769724&{}:{}&0.464101615138&{}:{}&0.464101615138&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.500000000000&{}:{}&0.500000000000&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.133974596216&{}:{}&0.000000000000&{}:{}&0.866025403784&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.133974596216&{}:{}&0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3a}}&{}\approx{}&-0.154700538379&{}:{}&0.577350269190&{}:{}&0.577350269190&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.366025403784&{}:{}&0.683012701892&{}:{}&0.683012701892&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.049038105677&{}:{}&0.366025403784&{}:{}&0.683012701892&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.049038105677&{}:{}&0.683012701892&{}:{}&0.366025403784&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.500000000000&{}:{}&0.500000000000&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.133974596216&{}:{}&0.000000000000&{}:{}&0.866025403784&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.133974596216&{}:{}&0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)