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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7c(332)

Malfatti circles

7c (332)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.000000000000&{}:{}&1.000000000000&{}:{}&-1.000000000000&, \\ P_{\mathbf{7c}}&{}\approx{}&-0.008106760913&{}:{}&-0.008106760913&{}:{}&1.016213521826&, \\ P^-_{\mathbf{7c}}&{}\approx{}&0.021497673936&{}:{}&0.021497673936&{}:{}&0.957004652127&, \\ P^+_{\mathbf{7c}}&{}\approx{}&-0.039558438908&{}:{}&-0.039558438908&{}:{}&1.079116877816&, \\ Q_{\mathbf{7c}}&{}\approx{}&0.118146029605&{}:{}&0.118146029605&{}:{}&0.763707940790&, \\ I^\prime_{\mathbf{7c}}&{}\approx{}&-0.108741129337&{}:{}&-0.108741129337&{}:{}&1.217482258674&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{7c}}\)
\(P^-_{\mathbf{7c}}\)
\(P^+_{\mathbf{7c}}\)
\(Q_{\mathbf{7c}}\)
\(I^\prime_{\mathbf{7c}}\)
7c (332)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7c}}&{}\approx{}&1.000000000000&{}:{}&-1.366025403784&{}:{}&1.366025403784&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-1.366025403784&{}:{}&1.000000000000&{}:{}&1.366025403784&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.032692070451&{}:{}&-0.032692070451&{}:{}&1.065384140902&, \\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.063855924532&{}:{}&-0.008555071705&{}:{}&1.072410996238&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.008555071705&{}:{}&-0.063855924532&{}:{}&1.072410996238&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.183012701892&{}:{}&-0.183012701892&{}:{}&1.366025403784&, \\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.030940107676&{}:{}&0.022649730810&{}:{}&1.008290376865&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.022649730810&{}:{}&-0.030940107676&{}:{}&1.008290376865&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.297086290221&{}:{}&0.297086290221&{}:{}&0.405827419558&, \\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.133974596216&{}:{}&0.866025403784&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.133974596216&{}:{}&0.000000000000&{}:{}&0.866025403784&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7c}}}{B^\prime_{\mathbf{7c}}}{C^\prime_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^*_{\mathbf{7c}}}{B^*_{\mathbf{7c}}}{C^*_{\mathbf{7c}}}\)
7c (332)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7c}}}}&{}\approx{}&-1.366025403784&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7c}}}}&{}\approx{}&-1.366025403784&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.032692070451&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.000000000000&{}:{}&1.000000000000&{}:{}&-1.000000000000&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&1.000000000000&{}:{}&-1.366025403784&{}:{}&1.366025403784&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-1.366025403784&{}:{}&1.000000000000&{}:{}&1.366025403784&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.032692070451&{}:{}&-0.032692070451&{}:{}&1.065384140902&. \end{alignedat} \]
7c (332)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7c}}&{}\approx{}&-0.008106760913&{}:{}&-0.008106760913&{}:{}&1.016213521826&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.063855924532&{}:{}&-0.008555071705&{}:{}&1.072410996238&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.008555071705&{}:{}&-0.063855924532&{}:{}&1.072410996238&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.183012701892&{}:{}&-0.183012701892&{}:{}&1.366025403784&. \end{alignedat} \]
7c (332)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7c}}&{}\approx{}&0.021497673936&{}:{}&0.021497673936&{}:{}&0.957004652127&,\\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.030940107676&{}:{}&0.022649730810&{}:{}&1.008290376865&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.022649730810&{}:{}&-0.030940107676&{}:{}&1.008290376865&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.297086290221&{}:{}&0.297086290221&{}:{}&0.405827419558&. \end{alignedat} \]
7c (332)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7c}}&{}\approx{}&-0.039558438908&{}:{}&-0.039558438908&{}:{}&1.079116877816&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&1.000000000000&{}:{}&-1.366025403784&{}:{}&1.366025403784&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-1.366025403784&{}:{}&1.000000000000&{}:{}&1.366025403784&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.032692070451&{}:{}&-0.032692070451&{}:{}&1.065384140902&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.063855924532&{}:{}&-0.008555071705&{}:{}&1.072410996238&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.008555071705&{}:{}&-0.063855924532&{}:{}&1.072410996238&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.183012701892&{}:{}&-0.183012701892&{}:{}&1.366025403784&, \end{alignedat} \]
7c (332)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7c}}&{}\approx{}&0.118146029605&{}:{}&0.118146029605&{}:{}&0.763707940790&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.133974596216&{}:{}&0.866025403784&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.133974596216&{}:{}&0.000000000000&{}:{}&0.866025403784&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7c}}&{}\approx{}&-0.108741129337&{}:{}&-0.108741129337&{}:{}&1.217482258674&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.063855924532&{}:{}&-0.008555071705&{}:{}&1.072410996238&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.008555071705&{}:{}&-0.063855924532&{}:{}&1.072410996238&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.183012701892&{}:{}&-0.183012701892&{}:{}&1.366025403784&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.133974596216&{}:{}&0.866025403784&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.133974596216&{}:{}&0.000000000000&{}:{}&0.866025403784&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)