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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7b (323)

Malfatti circles

7b (323)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&1.000000000000&{}:{}&-1.000000000000&{}:{}&1.000000000000&, \\ P_{\mathbf{7b}}&{}\approx{}&-0.008106760913&{}:{}&1.016213521826&{}:{}&-0.008106760913&, \\ P^-_{\mathbf{7b}}&{}\approx{}&0.021497673936&{}:{}&0.957004652127&{}:{}&0.021497673936&, \\ P^+_{\mathbf{7b}}&{}\approx{}&-0.039558438908&{}:{}&1.079116877816&{}:{}&-0.039558438908&, \\ Q_{\mathbf{7b}}&{}\approx{}&0.118146029605&{}:{}&0.763707940790&{}:{}&0.118146029605&, \\ I^\prime_{\mathbf{7b}}&{}\approx{}&-0.108741129337&{}:{}&1.217482258674&{}:{}&-0.108741129337&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{7b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{7b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{7b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{7b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{7b}}\) Radical center of the Malfatti circles
7b (323)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7b}}&{}\approx{}&1.000000000000&{}:{}&1.366025403784&{}:{}&-1.366025403784&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.032692070451&{}:{}&1.065384140902&{}:{}&-0.032692070451&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-1.366025403784&{}:{}&1.366025403784&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.063855924532&{}:{}&1.072410996238&{}:{}&-0.008555071705&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.183012701892&{}:{}&1.366025403784&{}:{}&-0.183012701892&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.008555071705&{}:{}&1.072410996238&{}:{}&-0.063855924532&, \\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.030940107676&{}:{}&1.008290376865&{}:{}&0.022649730810&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.297086290221&{}:{}&0.405827419558&{}:{}&0.297086290221&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.022649730810&{}:{}&1.008290376865&{}:{}&-0.030940107676&, \\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.866025403784&{}:{}&0.133974596216&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.500000000000&{}:{}&0.000000000000&{}:{}&0.500000000000&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.133974596216&{}:{}&0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7b}}}{B^\prime_{\mathbf{7b}}}{C^\prime_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^*_{\mathbf{7b}}}{B^*_{\mathbf{7b}}}{C^*_{\mathbf{7b}}}\)
7b (323)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7b}}}}&{}\approx{}&-1.366025403784&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7b}}}}&{}\approx{}&-0.032692070451&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7b}}}}&{}\approx{}&-1.366025403784&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&1.000000000000&{}:{}&-1.000000000000&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&1.000000000000&{}:{}&1.366025403784&{}:{}&-1.366025403784&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.032692070451&{}:{}&1.065384140902&{}:{}&-0.032692070451&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-1.366025403784&{}:{}&1.366025403784&{}:{}&1.000000000000&. \end{alignedat} \]
7b (323)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7b}}&{}\approx{}&-0.008106760913&{}:{}&1.016213521826&{}:{}&-0.008106760913&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.063855924532&{}:{}&1.072410996238&{}:{}&-0.008555071705&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.183012701892&{}:{}&1.366025403784&{}:{}&-0.183012701892&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.008555071705&{}:{}&1.072410996238&{}:{}&-0.063855924532&. \end{alignedat} \]
7b (323)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7b}}&{}\approx{}&0.021497673936&{}:{}&0.957004652127&{}:{}&0.021497673936&,\\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.030940107676&{}:{}&1.008290376865&{}:{}&0.022649730810&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.297086290221&{}:{}&0.405827419558&{}:{}&0.297086290221&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.022649730810&{}:{}&1.008290376865&{}:{}&-0.030940107676&. \end{alignedat} \]
7b (323)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7b}}&{}\approx{}&-0.039558438908&{}:{}&1.079116877816&{}:{}&-0.039558438908&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&1.000000000000&{}:{}&1.366025403784&{}:{}&-1.366025403784&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.032692070451&{}:{}&1.065384140902&{}:{}&-0.032692070451&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-1.366025403784&{}:{}&1.366025403784&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.063855924532&{}:{}&1.072410996238&{}:{}&-0.008555071705&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.183012701892&{}:{}&1.366025403784&{}:{}&-0.183012701892&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.008555071705&{}:{}&1.072410996238&{}:{}&-0.063855924532&, \end{alignedat} \]
7b (323)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7b}}&{}\approx{}&0.118146029605&{}:{}&0.763707940790&{}:{}&0.118146029605&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.866025403784&{}:{}&0.133974596216&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.500000000000&{}:{}&0.000000000000&{}:{}&0.500000000000&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.133974596216&{}:{}&0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7b}}&{}\approx{}&-0.108741129337&{}:{}&1.217482258674&{}:{}&-0.108741129337&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.063855924532&{}:{}&1.072410996238&{}:{}&-0.008555071705&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.183012701892&{}:{}&1.366025403784&{}:{}&-0.183012701892&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.008555071705&{}:{}&1.072410996238&{}:{}&-0.063855924532&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.866025403784&{}:{}&0.133974596216&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.500000000000&{}:{}&0.000000000000&{}:{}&0.500000000000&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.133974596216&{}:{}&0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)