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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7(222)

Malfatti circles

7 (222)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&, \\ P_{\mathbf{7}}&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&, \\ P^-_{\mathbf{7}}&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&, \\ P^+_{\mathbf{7}}&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&, \\ Q_{\mathbf{7}}&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&, \\ I^\prime_{\mathbf{7}}&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{7}}\)
\(P^-_{\mathbf{7}}\)
\(P^+_{\mathbf{7}}\)
\(Q_{\mathbf{7}}\)
\(I^\prime_{\mathbf{7}}\)
7 (222)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7}}&{}\approx{}&-0.577350269190&{}:{}&0.788675134595&{}:{}&0.788675134595&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.788675134595&{}:{}&-0.577350269190&{}:{}&0.788675134595&,\\C^\prime_{\mathbf{7}}&{}\approx{}&0.788675134595&{}:{}&0.788675134595&{}:{}&-0.577350269190&, \\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.788675134595&{}:{}&0.105662432703&{}:{}&0.105662432703&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.105662432703&{}:{}&0.788675134595&{}:{}&0.105662432703&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.105662432703&{}:{}&0.105662432703&{}:{}&0.788675134595&, \\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-2.154700538379&{}:{}&1.577350269190&{}:{}&1.577350269190&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&1.577350269190&{}:{}&-2.154700538379&{}:{}&1.577350269190&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&1.577350269190&{}:{}&1.577350269190&{}:{}&-2.154700538379&, \\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.500000000000&{}:{}&0.500000000000&,\\B^*_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&0.000000000000&{}:{}&0.500000000000&,\\C^*_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7}}}{B^\prime_{\mathbf{7}}}{C^\prime_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7}}}{B^{\prime\prime}_{\mathbf{7}}}{C^{\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7}}}{B^{\prime\prime\prime}_{\mathbf{7}}}{C^{\prime\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^*_{\mathbf{7}}}{B^*_{\mathbf{7}}}{C^*_{\mathbf{7}}}\)
7 (222)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7}}}}&{}\approx{}&2.366025403784&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7}}}}&{}\approx{}&2.366025403784&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7}}}}&{}\approx{}&2.366025403784&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&-0.577350269190&{}:{}&0.788675134595&{}:{}&0.788675134595&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.788675134595&{}:{}&-0.577350269190&{}:{}&0.788675134595&,\\C^\prime_{\mathbf{7}}&{}\approx{}&0.788675134595&{}:{}&0.788675134595&{}:{}&-0.577350269190&. \end{alignedat} \]
7 (222)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7}}&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.788675134595&{}:{}&0.105662432703&{}:{}&0.105662432703&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.105662432703&{}:{}&0.788675134595&{}:{}&0.105662432703&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.105662432703&{}:{}&0.105662432703&{}:{}&0.788675134595&. \end{alignedat} \]
7 (222)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7}}&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&,\\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-2.154700538379&{}:{}&1.577350269190&{}:{}&1.577350269190&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&1.577350269190&{}:{}&-2.154700538379&{}:{}&1.577350269190&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&1.577350269190&{}:{}&1.577350269190&{}:{}&-2.154700538379&. \end{alignedat} \]
7 (222)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7}}&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&-0.577350269190&{}:{}&0.788675134595&{}:{}&0.788675134595&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.788675134595&{}:{}&-0.577350269190&{}:{}&0.788675134595&,\\C^\prime_{\mathbf{7}}&{}\approx{}&0.788675134595&{}:{}&0.788675134595&{}:{}&-0.577350269190&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.788675134595&{}:{}&0.105662432703&{}:{}&0.105662432703&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.105662432703&{}:{}&0.788675134595&{}:{}&0.105662432703&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.105662432703&{}:{}&0.105662432703&{}:{}&0.788675134595&, \end{alignedat} \]
7 (222)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7}}&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.500000000000&{}:{}&0.500000000000&,\\B^*_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&0.000000000000&{}:{}&0.500000000000&,\\C^*_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7}}&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.788675134595&{}:{}&0.105662432703&{}:{}&0.105662432703&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.105662432703&{}:{}&0.788675134595&{}:{}&0.105662432703&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.105662432703&{}:{}&0.105662432703&{}:{}&0.788675134595&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.500000000000&{}:{}&0.500000000000&,\\B^*_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&0.000000000000&{}:{}&0.500000000000&,\\C^*_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)