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,
\[ \begin{alignedat}{4} I&{}\approx{}&0.214285714286&{}:{}&0.321428571429&{}:{}&0.464285714286&, \\ P_{\mathbf{3}}&{}\approx{}&0.000464543439&{}:{}&0.983182621155&{}:{}&0.016352835406&, \\ P^-_{\mathbf{3}}&{}\approx{}&-0.018877135280&{}:{}&1.043043083848&{}:{}&-0.024165948568&, \\ P^+_{\mathbf{3}}&{}\approx{}&0.016843102738&{}:{}&0.932492701622&{}:{}&0.050664195641&, \\ Q_{\mathbf{3}}&{}\approx{}&-0.013518355959&{}:{}&0.786661327558&{}:{}&0.226857028401&, \\ I^\prime_{\mathbf{3}}&{}\approx{}&0.015133729456&{}:{}&0.852698554716&{}:{}&0.132167715827&, \end{alignedat} \]
\(I\) Incenter
\(P_{\mathbf{3}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{3}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{3}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{3}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{3}}\) Radical center of the Malfatti circles
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3}}&{}\approx{}&-0.361655667145&{}:{}&0.557040954741&{}:{}&0.804614712404&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.017550946010&{}:{}&0.944422004303&{}:{}&0.038027049688&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.797903076547&{}:{}&1.196854614821&{}:{}&-0.994757691368&, \\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.034346397050&{}:{}&0.949855089425&{}:{}&0.015798513524&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006622650490&{}:{}&0.760247130897&{}:{}&0.233130218614&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000437969453&{}:{}&0.926940128751&{}:{}&0.072621901796&, \\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.018669393459&{}:{}&1.004606018408&{}:{}&-0.023275411867&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&1.351500958093&{}:{}&-2.081652390445&{}:{}&1.730151432352&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.017705658692&{}:{}&0.978313953348&{}:{}&0.039391705344&, \\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.776168801416&{}:{}&0.223831198584&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.063365707697&{}:{}&0.000000000000&{}:{}&1.063365707697&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.017484936752&{}:{}&1.017484936752&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3}}}{B^\prime_{\mathbf{3}}}{C^\prime_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3}}}{B^{\prime\prime}_{\mathbf{3}}}{C^{\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3}}}{B^{\prime\prime\prime}_{\mathbf{3}}}{C^{\prime\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^*_{\mathbf{3}}}{B^*_{\mathbf{3}}}{C^*_{\mathbf{3}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3}}}}&{}\approx{}&1.733016303638&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3}}}}&{}\approx{}&0.081904414712&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3}}}}&{}\approx{}&3.723547690553&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.214285714286&{}:{}&0.321428571429&{}:{}&0.464285714286&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.361655667145&{}:{}&0.557040954741&{}:{}&0.804614712404&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.017550946010&{}:{}&0.944422004303&{}:{}&0.038027049688&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.797903076547&{}:{}&1.196854614821&{}:{}&-0.994757691368&. \end{alignedat} \]
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First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3}}&{}\approx{}&0.000464543439&{}:{}&0.983182621155&{}:{}&0.016352835406&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.034346397050&{}:{}&0.949855089425&{}:{}&0.015798513524&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006622650490&{}:{}&0.760247130897&{}:{}&0.233130218614&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000437969453&{}:{}&0.926940128751&{}:{}&0.072621901796&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3}}&{}\approx{}&-0.018877135280&{}:{}&1.043043083848&{}:{}&-0.024165948568&,\\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.018669393459&{}:{}&1.004606018408&{}:{}&-0.023275411867&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&1.351500958093&{}:{}&-2.081652390445&{}:{}&1.730151432352&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.017705658692&{}:{}&0.978313953348&{}:{}&0.039391705344&. \end{alignedat} \]
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Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3}}&{}\approx{}&0.016843102738&{}:{}&0.932492701622&{}:{}&0.050664195641&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.361655667145&{}:{}&0.557040954741&{}:{}&0.804614712404&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.017550946010&{}:{}&0.944422004303&{}:{}&0.038027049688&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.797903076547&{}:{}&1.196854614821&{}:{}&-0.994757691368&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.034346397050&{}:{}&0.949855089425&{}:{}&0.015798513524&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006622650490&{}:{}&0.760247130897&{}:{}&0.233130218614&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000437969453&{}:{}&0.926940128751&{}:{}&0.072621901796&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3}}&{}\approx{}&-0.013518355959&{}:{}&0.786661327558&{}:{}&0.226857028401&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.776168801416&{}:{}&0.223831198584&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.063365707697&{}:{}&0.000000000000&{}:{}&1.063365707697&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.017484936752&{}:{}&1.017484936752&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3}}&{}\approx{}&0.015133729456&{}:{}&0.852698554716&{}:{}&0.132167715827&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.034346397050&{}:{}&0.949855089425&{}:{}&0.015798513524&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006622650490&{}:{}&0.760247130897&{}:{}&0.233130218614&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000437969453&{}:{}&0.926940128751&{}:{}&0.072621901796&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.776168801416&{}:{}&0.223831198584&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.063365707697&{}:{}&0.000000000000&{}:{}&1.063365707697&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.017484936752&{}:{}&1.017484936752&{}:{}&0.000000000000&. \end{alignedat} \]
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