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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7a(233)

Malfatti circles

7a (233)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-1.000000000000&{}:{}&1.000000000000&{}:{}&1.000000000000&, \\ P_{\mathbf{7a}}&{}\approx{}&1.016213521826&{}:{}&-0.008106760913&{}:{}&-0.008106760913&, \\ P^-_{\mathbf{7a}}&{}\approx{}&0.957004652127&{}:{}&0.021497673936&{}:{}&0.021497673936&, \\ P^+_{\mathbf{7a}}&{}\approx{}&1.079116877816&{}:{}&-0.039558438908&{}:{}&-0.039558438908&, \\ Q_{\mathbf{7a}}&{}\approx{}&0.763707940790&{}:{}&0.118146029605&{}:{}&0.118146029605&, \\ I^\prime_{\mathbf{7a}}&{}\approx{}&1.217482258674&{}:{}&-0.108741129337&{}:{}&-0.108741129337&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{7a}}\)
\(P^-_{\mathbf{7a}}\)
\(P^+_{\mathbf{7a}}\)
\(Q_{\mathbf{7a}}\)
\(I^\prime_{\mathbf{7a}}\)
7a (233)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7a}}&{}\approx{}&1.065384140902&{}:{}&-0.032692070451&{}:{}&-0.032692070451&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&1.366025403784&{}:{}&1.000000000000&{}:{}&-1.366025403784&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&1.366025403784&{}:{}&-1.366025403784&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.366025403784&{}:{}&-0.183012701892&{}:{}&-0.183012701892&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.072410996238&{}:{}&-0.063855924532&{}:{}&-0.008555071705&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.072410996238&{}:{}&-0.008555071705&{}:{}&-0.063855924532&, \\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.405827419558&{}:{}&0.297086290221&{}:{}&0.297086290221&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.008290376865&{}:{}&-0.030940107676&{}:{}&0.022649730810&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.008290376865&{}:{}&0.022649730810&{}:{}&-0.030940107676&, \\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.500000000000&{}:{}&0.500000000000&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.866025403784&{}:{}&0.000000000000&{}:{}&0.133974596216&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.866025403784&{}:{}&0.133974596216&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7a}}}{B^\prime_{\mathbf{7a}}}{C^\prime_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^*_{\mathbf{7a}}}{B^*_{\mathbf{7a}}}{C^*_{\mathbf{7a}}}\)
7a (233)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7a}}}}&{}\approx{}&-0.032692070451&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7a}}}}&{}\approx{}&-1.366025403784&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7a}}}}&{}\approx{}&-1.366025403784&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-1.000000000000&{}:{}&1.000000000000&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.065384140902&{}:{}&-0.032692070451&{}:{}&-0.032692070451&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&1.366025403784&{}:{}&1.000000000000&{}:{}&-1.366025403784&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&1.366025403784&{}:{}&-1.366025403784&{}:{}&1.000000000000&. \end{alignedat} \]
7a (233)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7a}}&{}\approx{}&1.016213521826&{}:{}&-0.008106760913&{}:{}&-0.008106760913&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.366025403784&{}:{}&-0.183012701892&{}:{}&-0.183012701892&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.072410996238&{}:{}&-0.063855924532&{}:{}&-0.008555071705&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.072410996238&{}:{}&-0.008555071705&{}:{}&-0.063855924532&. \end{alignedat} \]
7a (233)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7a}}&{}\approx{}&0.957004652127&{}:{}&0.021497673936&{}:{}&0.021497673936&,\\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.405827419558&{}:{}&0.297086290221&{}:{}&0.297086290221&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.008290376865&{}:{}&-0.030940107676&{}:{}&0.022649730810&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.008290376865&{}:{}&0.022649730810&{}:{}&-0.030940107676&. \end{alignedat} \]
7a (233)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7a}}&{}\approx{}&1.079116877816&{}:{}&-0.039558438908&{}:{}&-0.039558438908&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.065384140902&{}:{}&-0.032692070451&{}:{}&-0.032692070451&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&1.366025403784&{}:{}&1.000000000000&{}:{}&-1.366025403784&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&1.366025403784&{}:{}&-1.366025403784&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.366025403784&{}:{}&-0.183012701892&{}:{}&-0.183012701892&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.072410996238&{}:{}&-0.063855924532&{}:{}&-0.008555071705&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.072410996238&{}:{}&-0.008555071705&{}:{}&-0.063855924532&, \end{alignedat} \]
7a (233)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7a}}&{}\approx{}&0.763707940790&{}:{}&0.118146029605&{}:{}&0.118146029605&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.500000000000&{}:{}&0.500000000000&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.866025403784&{}:{}&0.000000000000&{}:{}&0.133974596216&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.866025403784&{}:{}&0.133974596216&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7a}}&{}\approx{}&1.217482258674&{}:{}&-0.108741129337&{}:{}&-0.108741129337&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.366025403784&{}:{}&-0.183012701892&{}:{}&-0.183012701892&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.072410996238&{}:{}&-0.063855924532&{}:{}&-0.008555071705&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.072410996238&{}:{}&-0.008555071705&{}:{}&-0.063855924532&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.500000000000&{}:{}&0.500000000000&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.866025403784&{}:{}&0.000000000000&{}:{}&0.133974596216&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.866025403784&{}:{}&0.133974596216&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)