Derousseau's Generalization of the Malfatti circles

Test Case in ETC

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

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


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\(\mathbf{3c}\) \((132)\)

Malfatti circles

3c (132)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&4.500000000000&{}:{}&-6.500000000000&, \\ P_{\mathbf{3c}}&{}\approx{}&-0.135700778741&{}:{}&-0.641504235939&{}:{}&1.777205014680&, \\ P^-_{\mathbf{3c}}&{}\approx{}&0.101411483793&{}:{}&-0.252719148256&{}:{}&1.151307664463&, \\ P^+_{\mathbf{3c}}&{}\approx{}&-0.415061943527&{}:{}&-1.099563442940&{}:{}&2.514625386466&, \\ Q_{\mathbf{3c}}&{}\approx{}&0.692607399846&{}:{}&-1.190517353287&{}:{}&1.497909953441&, \\ I^\prime_{\mathbf{3c}}&{}\approx{}&-0.600964112749&{}:{}&-1.600305000045&{}:{}&3.201269112794&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{3c}}\)
\(P^-_{\mathbf{3c}}\)
\(P^+_{\mathbf{3c}}\)
\(Q_{\mathbf{3c}}\)
\(I^\prime_{\mathbf{3c}}\)
3c (132)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3c}}&{}\approx{}&0.401398387271&{}:{}&-1.346853628640&{}:{}&1.945455241369&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-18.998725976367&{}:{}&-21.165180305761&{}:{}&41.163906282128&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.388323619932&{}:{}&-0.582485429898&{}:{}&1.970809049831&, \\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.761090954928&{}:{}&-0.994757887471&{}:{}&2.755848842400&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.149899191198&{}:{}&-0.813255307735&{}:{}&1.963154498933&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.474097613890&{}:{}&-2.241222418773&{}:{}&3.715320032663&, \\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.337707757521&{}:{}&-0.376217099372&{}:{}&1.713924856893&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.111142854050&{}:{}&-0.372929142272&{}:{}&1.261786288222&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.298091278401&{}:{}&-0.742848552872&{}:{}&1.444757274472&, \\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-3.872953846944&{}:{}&4.872953846944&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.316184393064&{}:{}&0.000000000000&{}:{}&0.683815606936&,\\C^*_{\mathbf{3c}}&{}\approx{}&-1.391029432250&{}:{}&2.391029432250&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3c}}}{B^\prime_{\mathbf{3c}}}{C^\prime_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^*_{\mathbf{3c}}}{B^*_{\mathbf{3c}}}{C^*_{\mathbf{3c}}}\)
3c (132)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.299300806364&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3c}}}}&{}\approx{}&-6.332908658789&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.129441206644&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&4.500000000000&{}:{}&-6.500000000000&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.401398387271&{}:{}&-1.346853628640&{}:{}&1.945455241369&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-18.998725976367&{}:{}&-21.165180305761&{}:{}&41.163906282128&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.388323619932&{}:{}&-0.582485429898&{}:{}&1.970809049831&. \end{alignedat} \]
3c (132)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3c}}&{}\approx{}&-0.135700778741&{}:{}&-0.641504235939&{}:{}&1.777205014680&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.761090954928&{}:{}&-0.994757887471&{}:{}&2.755848842400&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.149899191198&{}:{}&-0.813255307735&{}:{}&1.963154498933&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.474097613890&{}:{}&-2.241222418773&{}:{}&3.715320032663&. \end{alignedat} \]
3c (132)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3c}}&{}\approx{}&0.101411483793&{}:{}&-0.252719148256&{}:{}&1.151307664463&,\\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.337707757521&{}:{}&-0.376217099372&{}:{}&1.713924856893&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.111142854050&{}:{}&-0.372929142272&{}:{}&1.261786288222&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.298091278401&{}:{}&-0.742848552872&{}:{}&1.444757274472&. \end{alignedat} \]
3c (132)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3c}}&{}\approx{}&-0.415061943527&{}:{}&-1.099563442940&{}:{}&2.514625386466&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.401398387271&{}:{}&-1.346853628640&{}:{}&1.945455241369&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-18.998725976367&{}:{}&-21.165180305761&{}:{}&41.163906282128&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.388323619932&{}:{}&-0.582485429898&{}:{}&1.970809049831&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.761090954928&{}:{}&-0.994757887471&{}:{}&2.755848842400&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.149899191198&{}:{}&-0.813255307735&{}:{}&1.963154498933&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.474097613890&{}:{}&-2.241222418773&{}:{}&3.715320032663&, \end{alignedat} \]
3c (132)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3c}}&{}\approx{}&0.692607399846&{}:{}&-1.190517353287&{}:{}&1.497909953441&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-3.872953846944&{}:{}&4.872953846944&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.316184393064&{}:{}&0.000000000000&{}:{}&0.683815606936&,\\C^*_{\mathbf{3c}}&{}\approx{}&-1.391029432250&{}:{}&2.391029432250&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3c}}&{}\approx{}&-0.600964112749&{}:{}&-1.600305000045&{}:{}&3.201269112794&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.761090954928&{}:{}&-0.994757887471&{}:{}&2.755848842400&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.149899191198&{}:{}&-0.813255307735&{}:{}&1.963154498933&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.474097613890&{}:{}&-2.241222418773&{}:{}&3.715320032663&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-3.872953846944&{}:{}&4.872953846944&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.316184393064&{}:{}&0.000000000000&{}:{}&0.683815606936&,\\C^*_{\mathbf{3c}}&{}\approx{}&-1.391029432250&{}:{}&2.391029432250&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)

Hiroyasu Kamo