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,
\[ \begin{alignedat}{4} I&{}\approx{}&0.214285714286&{}:{}&0.321428571429&{}:{}&0.464285714286&, \\ P_{\mathbf{6}}&{}\approx{}&0.798246585324&{}:{}&0.201399099006&{}:{}&0.000354315670&, \\ P^-_{\mathbf{6}}&{}\approx{}&1.396778541197&{}:{}&0.078374632035&{}:{}&-0.475153173232&, \\ P^+_{\mathbf{6}}&{}\approx{}&0.602000443398&{}:{}&0.241736255224&{}:{}&0.156263301378&, \\ Q_{\mathbf{6}}&{}\approx{}&0.634610079081&{}:{}&0.403206183454&{}:{}&-0.037816262535&, \\ I^\prime_{\mathbf{6}}&{}\approx{}&0.607460835328&{}:{}&0.373700896310&{}:{}&0.018838268363&, \end{alignedat} \]
\(I\) Incenter
\(P_{\mathbf{6}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{6}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{6}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{6}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{6}}\) Radical center of the Malfatti circles
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6}}&{}\approx{}&0.634294423249&{}:{}&0.149606826853&{}:{}&0.216098749898&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.283807833766&{}:{}&0.101275193075&{}:{}&0.614916973159&,\\C^\prime_{\mathbf{6}}&{}\approx{}&2.970885091952&{}:{}&4.456327637927&{}:{}&-6.427212729879&, \\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.518119823625&{}:{}&0.481033907190&{}:{}&0.000846269184&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.710189970238&{}:{}&0.289494799556&{}:{}&0.000315230206&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.543153291088&{}:{}&0.137038586144&{}:{}&0.319808122768&, \\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&-1.239839937101&{}:{}&-0.442429750252&{}:{}&2.682269687353&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.116463320802&{}:{}&0.263332812965&{}:{}&-0.379796133767&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.715920804132&{}:{}&0.040171027786&{}:{}&0.243908168081&, \\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.103495636770&{}:{}&-0.103495636770&,\\B^*_{\mathbf{6}}&{}\approx{}&1.063365707697&{}:{}&0.000000000000&{}:{}&-0.063365707697&,\\C^*_{\mathbf{6}}&{}\approx{}&0.611485965281&{}:{}&0.388514034719&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6}}}{B^\prime_{\mathbf{6}}}{C^\prime_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6}}}{B^{\prime\prime}_{\mathbf{6}}}{C^{\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6}}}{B^{\prime\prime\prime}_{\mathbf{6}}}{C^{\prime\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^*_{\mathbf{6}}}{B^*_{\mathbf{6}}}{C^*_{\mathbf{6}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6}}}}&{}\approx{}&0.465443461319&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6}}}}&{}\approx{}&1.324436557573&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6}}}}&{}\approx{}&13.864130429108&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.214285714286&{}:{}&0.321428571429&{}:{}&0.464285714286&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.634294423249&{}:{}&0.149606826853&{}:{}&0.216098749898&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.283807833766&{}:{}&0.101275193075&{}:{}&0.614916973159&,\\C^\prime_{\mathbf{6}}&{}\approx{}&2.970885091952&{}:{}&4.456327637927&{}:{}&-6.427212729879&. \end{alignedat} \]
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First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6}}&{}\approx{}&0.798246585324&{}:{}&0.201399099006&{}:{}&0.000354315670&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.518119823625&{}:{}&0.481033907190&{}:{}&0.000846269184&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.710189970238&{}:{}&0.289494799556&{}:{}&0.000315230206&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.543153291088&{}:{}&0.137038586144&{}:{}&0.319808122768&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6}}&{}\approx{}&1.396778541197&{}:{}&0.078374632035&{}:{}&-0.475153173232&,\\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&-1.239839937101&{}:{}&-0.442429750252&{}:{}&2.682269687353&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.116463320802&{}:{}&0.263332812965&{}:{}&-0.379796133767&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.715920804132&{}:{}&0.040171027786&{}:{}&0.243908168081&. \end{alignedat} \]
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Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6}}&{}\approx{}&0.602000443398&{}:{}&0.241736255224&{}:{}&0.156263301378&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.634294423249&{}:{}&0.149606826853&{}:{}&0.216098749898&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.283807833766&{}:{}&0.101275193075&{}:{}&0.614916973159&,\\C^\prime_{\mathbf{6}}&{}\approx{}&2.970885091952&{}:{}&4.456327637927&{}:{}&-6.427212729879&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.518119823625&{}:{}&0.481033907190&{}:{}&0.000846269184&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.710189970238&{}:{}&0.289494799556&{}:{}&0.000315230206&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.543153291088&{}:{}&0.137038586144&{}:{}&0.319808122768&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6}}&{}\approx{}&0.634610079081&{}:{}&0.403206183454&{}:{}&-0.037816262535&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.103495636770&{}:{}&-0.103495636770&,\\B^*_{\mathbf{6}}&{}\approx{}&1.063365707697&{}:{}&0.000000000000&{}:{}&-0.063365707697&,\\C^*_{\mathbf{6}}&{}\approx{}&0.611485965281&{}:{}&0.388514034719&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6}}&{}\approx{}&0.607460835328&{}:{}&0.373700896310&{}:{}&0.018838268363&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.518119823625&{}:{}&0.481033907190&{}:{}&0.000846269184&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.710189970238&{}:{}&0.289494799556&{}:{}&0.000315230206&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.543153291088&{}:{}&0.137038586144&{}:{}&0.319808122768&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.103495636770&{}:{}&-0.103495636770&,\\B^*_{\mathbf{6}}&{}\approx{}&1.063365707697&{}:{}&0.000000000000&{}:{}&-0.063365707697&,\\C^*_{\mathbf{6}}&{}\approx{}&0.611485965281&{}:{}&0.388514034719&{}:{}&0.000000000000&. \end{alignedat} \]
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