Derousseau's Generalization of the Malfatti circles

Test Case in ETC

The test case used in “Encyclopedis of Triangle Centers” to search for triangle centers.

\(a:b:c=6:9:13\).

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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.214285714286&{}:{}&0.321428571429&{}:{}&0.464285714286&, \\ P_{\mathbf{2}}&{}\approx{}&0.000471437338&{}:{}&0.997773208815&{}:{}&0.001755353847&, \\ P^-_{\mathbf{2}}&{}\approx{}&-0.005547399205&{}:{}&1.016812198527&{}:{}&-0.011264799322&, \\ P^+_{\mathbf{2}}&{}\approx{}&0.006169476178&{}:{}&0.979748977372&{}:{}&0.014081546449&, \\ Q_{\mathbf{2}}&{}\approx{}&-0.019329530417&{}:{}&1.124825689246&{}:{}&-0.105496158829&, \\ I^\prime_{\mathbf{2}}&{}\approx{}&0.016615557506&{}:{}&0.936191036845&{}:{}&0.047193405649&, \end{alignedat} \]
\(I\) Incenter
\(P_{\mathbf{2}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{2}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{2}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{2}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{2}}\) Radical center of the Malfatti circles
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2}}&{}\approx{}&-0.334779082061&{}:{}&0.546045988116&{}:{}&0.788733093945&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.005822981610&{}:{}&0.981560558236&{}:{}&0.012616460154&,\\C^\prime_{\mathbf{2}}&{}\approx{}&1.580960831172&{}:{}&2.371441246757&{}:{}&-2.952402077929&, \\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.011603226314&{}:{}&0.986660969284&{}:{}&0.001735804402&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.023776760580&{}:{}&0.887692644112&{}:{}&0.088530595308&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.000460537290&{}:{}&0.974703809901&{}:{}&0.024835652809&, \\ A^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&0.005963058155&{}:{}&1.005172793499&{}:{}&-0.011135851654&,\\B^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&0.714499290504&{}:{}&-1.165393813405&{}:{}&1.450894522901&,\\C^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&-0.005415613683&{}:{}&0.992656531755&{}:{}&0.012759081928&, \\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.103495636770&{}:{}&-0.103495636770&,\\B^*_{\mathbf{2}}&{}\approx{}&0.154852182538&{}:{}&0.000000000000&{}:{}&0.845147817462&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.017484936752&{}:{}&1.017484936752&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2}}}{B^\prime_{\mathbf{2}}}{C^\prime_{\mathbf{2}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2}}}{B^{\prime\prime}_{\mathbf{2}}}{C^{\prime\prime}_{\mathbf{2}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2}}}{B^{\prime\prime\prime}_{\mathbf{2}}}{C^{\prime\prime\prime}_{\mathbf{2}}}\)
\(\triangle{A^*_{\mathbf{2}}}{B^*_{\mathbf{2}}}{C^*_{\mathbf{2}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2}}}}&{}\approx{}&1.698809740805&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2}}}}&{}\approx{}&0.027173914178&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2}}}}&{}\approx{}&7.377817212134&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.214285714286&{}:{}&0.321428571429&{}:{}&0.464285714286&,\\ A^\prime_{\mathbf{2}}&{}\approx{}&-0.334779082061&{}:{}&0.546045988116&{}:{}&0.788733093945&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.005822981610&{}:{}&0.981560558236&{}:{}&0.012616460154&,\\C^\prime_{\mathbf{2}}&{}\approx{}&1.580960831172&{}:{}&2.371441246757&{}:{}&-2.952402077929&. \end{alignedat} \]
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First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2}}&{}\approx{}&0.000471437338&{}:{}&0.997773208815&{}:{}&0.001755353847&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.011603226314&{}:{}&0.986660969284&{}:{}&0.001735804402&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.023776760580&{}:{}&0.887692644112&{}:{}&0.088530595308&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.000460537290&{}:{}&0.974703809901&{}:{}&0.024835652809&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2}}&{}\approx{}&-0.005547399205&{}:{}&1.016812198527&{}:{}&-0.011264799322&,\\ A^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&0.005963058155&{}:{}&1.005172793499&{}:{}&-0.011135851654&,\\B^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&0.714499290504&{}:{}&-1.165393813405&{}:{}&1.450894522901&,\\C^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&-0.005415613683&{}:{}&0.992656531755&{}:{}&0.012759081928&. \end{alignedat} \]
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Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2}}&{}\approx{}&0.006169476178&{}:{}&0.979748977372&{}:{}&0.014081546449&,\\ A^\prime_{\mathbf{2}}&{}\approx{}&-0.334779082061&{}:{}&0.546045988116&{}:{}&0.788733093945&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.005822981610&{}:{}&0.981560558236&{}:{}&0.012616460154&,\\C^\prime_{\mathbf{2}}&{}\approx{}&1.580960831172&{}:{}&2.371441246757&{}:{}&-2.952402077929&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.011603226314&{}:{}&0.986660969284&{}:{}&0.001735804402&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.023776760580&{}:{}&0.887692644112&{}:{}&0.088530595308&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.000460537290&{}:{}&0.974703809901&{}:{}&0.024835652809&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2}}&{}\approx{}&-0.019329530417&{}:{}&1.124825689246&{}:{}&-0.105496158829&,\\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.103495636770&{}:{}&-0.103495636770&,\\B^*_{\mathbf{2}}&{}\approx{}&0.154852182538&{}:{}&0.000000000000&{}:{}&0.845147817462&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.017484936752&{}:{}&1.017484936752&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2}}&{}\approx{}&0.016615557506&{}:{}&0.936191036845&{}:{}&0.047193405649&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.011603226314&{}:{}&0.986660969284&{}:{}&0.001735804402&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.023776760580&{}:{}&0.887692644112&{}:{}&0.088530595308&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.000460537290&{}:{}&0.974703809901&{}:{}&0.024835652809&,\\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.103495636770&{}:{}&-0.103495636770&,\\B^*_{\mathbf{2}}&{}\approx{}&0.154852182538&{}:{}&0.000000000000&{}:{}&0.845147817462&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.017484936752&{}:{}&1.017484936752&{}:{}&0.000000000000&. \end{alignedat} \]
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