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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\(\mathbf{3b}\) \((123)\)

Malfatti circles

3b (123)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-0.900000000000&{}:{}&1.300000000000&, \\ P_{\mathbf{3b}}&{}\approx{}&-0.001602837804&{}:{}&1.262077123995&{}:{}&-0.260474286191&, \\ P^-_{\mathbf{3b}}&{}\approx{}&0.106733861837&{}:{}&0.872730057686&{}:{}&0.020536080478&, \\ P^+_{\mathbf{3b}}&{}\approx{}&-0.170921313919&{}:{}&1.870584233458&{}:{}&-0.699662919539&, \\ Q_{\mathbf{3b}}&{}\approx{}&0.250470958986&{}:{}&1.878414879051&{}:{}&-1.128885838037&, \\ I^\prime_{\mathbf{3b}}&{}\approx{}&-0.068479950910&{}:{}&2.353462208093&{}:{}&-1.284982257183&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{3b}}\)
\(P^-_{\mathbf{3b}}\)
\(P^+_{\mathbf{3b}}\)
\(Q_{\mathbf{3b}}\)
\(I^\prime_{\mathbf{3b}}\)
3b (123)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3b}}&{}\approx{}&3.075742341626&{}:{}&4.670420268659&{}:{}&-6.746162610286&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.203642231078&{}:{}&1.644867065080&{}:{}&-0.441224834002&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-1.449140290046&{}:{}&2.173710435069&{}:{}&0.275429854977&, \\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.357102229771&{}:{}&1.710026783542&{}:{}&-0.352924553771&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.012027964215&{}:{}&2.966670776352&{}:{}&-1.954642812137&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.002324864380&{}:{}&1.830602037678&{}:{}&-0.828277173298&, \\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.137602614501&{}:{}&1.111449267985&{}:{}&0.026153346517&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.368884744243&{}:{}&0.560140153157&{}:{}&0.070975102600&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.143197773043&{}:{}&1.170884277753&{}:{}&-0.314082050796&, \\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&2.506126882702&{}:{}&-1.506126882702&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.285139704438&{}:{}&0.000000000000&{}:{}&1.285139704438&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.117653541825&{}:{}&0.882346458175&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3b}}}{B^\prime_{\mathbf{3b}}}{C^\prime_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^*_{\mathbf{3b}}}{B^*_{\mathbf{3b}}}{C^*_{\mathbf{3b}}}\)
3b (123)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3b}}}}&{}\approx{}&-5.189355854066&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3b}}}}&{}\approx{}&-0.339403718463&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3b}}}}&{}\approx{}&-2.415233816743&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-0.900000000000&{}:{}&1.300000000000&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&3.075742341626&{}:{}&4.670420268659&{}:{}&-6.746162610286&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.203642231078&{}:{}&1.644867065080&{}:{}&-0.441224834002&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-1.449140290046&{}:{}&2.173710435069&{}:{}&0.275429854977&. \end{alignedat} \]
3b (123)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3b}}&{}\approx{}&-0.001602837804&{}:{}&1.262077123995&{}:{}&-0.260474286191&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.357102229771&{}:{}&1.710026783542&{}:{}&-0.352924553771&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.012027964215&{}:{}&2.966670776352&{}:{}&-1.954642812137&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.002324864380&{}:{}&1.830602037678&{}:{}&-0.828277173298&. \end{alignedat} \]
3b (123)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3b}}&{}\approx{}&0.106733861837&{}:{}&0.872730057686&{}:{}&0.020536080478&,\\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.137602614501&{}:{}&1.111449267985&{}:{}&0.026153346517&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.368884744243&{}:{}&0.560140153157&{}:{}&0.070975102600&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.143197773043&{}:{}&1.170884277753&{}:{}&-0.314082050796&. \end{alignedat} \]
3b (123)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3b}}&{}\approx{}&-0.170921313919&{}:{}&1.870584233458&{}:{}&-0.699662919539&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&3.075742341626&{}:{}&4.670420268659&{}:{}&-6.746162610286&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.203642231078&{}:{}&1.644867065080&{}:{}&-0.441224834002&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-1.449140290046&{}:{}&2.173710435069&{}:{}&0.275429854977&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.357102229771&{}:{}&1.710026783542&{}:{}&-0.352924553771&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.012027964215&{}:{}&2.966670776352&{}:{}&-1.954642812137&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.002324864380&{}:{}&1.830602037678&{}:{}&-0.828277173298&, \end{alignedat} \]
3b (123)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3b}}&{}\approx{}&0.250470958986&{}:{}&1.878414879051&{}:{}&-1.128885838037&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&2.506126882702&{}:{}&-1.506126882702&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.285139704438&{}:{}&0.000000000000&{}:{}&1.285139704438&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.117653541825&{}:{}&0.882346458175&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3b}}&{}\approx{}&-0.068479950910&{}:{}&2.353462208093&{}:{}&-1.284982257183&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.357102229771&{}:{}&1.710026783542&{}:{}&-0.352924553771&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.012027964215&{}:{}&2.966670776352&{}:{}&-1.954642812137&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.002324864380&{}:{}&1.830602037678&{}:{}&-0.828277173298&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&2.506126882702&{}:{}&-1.506126882702&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.285139704438&{}:{}&0.000000000000&{}:{}&1.285139704438&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.117653541825&{}:{}&0.882346458175&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)

Hiroyasu Kamo