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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\(\mathbf{7a}\) \((233)\)

Malfatti circles

7a (233)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.375000000000&{}:{}&0.562500000000&{}:{}&0.812500000000&, \\ P_{\mathbf{7a}}&{}\approx{}&1.000694018324&{}:{}&-0.000401806075&{}:{}&-0.000292212249&, \\ P^-_{\mathbf{7a}}&{}\approx{}&0.984741086278&{}:{}&0.006125760478&{}:{}&0.009133153245&, \\ P^+_{\mathbf{7a}}&{}\approx{}&1.017025724421&{}:{}&-0.007084358108&{}:{}&-0.009941366313&, \\ Q_{\mathbf{7a}}&{}\approx{}&0.906149988922&{}:{}&0.067939399643&{}:{}&0.025910611434&, \\ I^\prime_{\mathbf{7a}}&{}\approx{}&1.052293625377&{}:{}&-0.025824990271&{}:{}&-0.026468635106&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{7a}}\)
\(P^-_{\mathbf{7a}}\)
\(P^+_{\mathbf{7a}}\)
\(Q_{\mathbf{7a}}\)
\(I^\prime_{\mathbf{7a}}\)
7a (233)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7a}}&{}\approx{}&1.016458450166&{}:{}&-0.006733002341&{}:{}&-0.009725447825&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&4.132129528091&{}:{}&5.820817782772&{}:{}&-8.952947310863&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.624902704110&{}:{}&-0.937354056165&{}:{}&1.312451352055&, \\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.085625991078&{}:{}&-0.049573681585&{}:{}&-0.036052309493&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.013665968550&{}:{}&-0.013369968368&{}:{}&-0.000296000183&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.019839282796&{}:{}&-0.000409493424&{}:{}&-0.019429789372&, \\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.221782513776&{}:{}&0.312418957663&{}:{}&0.465798528560&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.997356314757&{}:{}&-0.006606470142&{}:{}&0.009250155385&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.003358534485&{}:{}&0.006241573690&{}:{}&-0.009600108175&, \\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.723914668343&{}:{}&0.276085331657&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.972200722330&{}:{}&0.000000000000&{}:{}&0.027799277670&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.930253423925&{}:{}&0.069746576075&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7a}}}{B^\prime_{\mathbf{7a}}}{C^\prime_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^*_{\mathbf{7a}}}{B^*_{\mathbf{7a}}}{C^*_{\mathbf{7a}}}\)
7a (233)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7a}}}}&{}\approx{}&-0.011969781939&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7a}}}}&{}\approx{}&-11.019012074908&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7a}}}}&{}\approx{}&-1.666407210959&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.375000000000&{}:{}&0.562500000000&{}:{}&0.812500000000&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.016458450166&{}:{}&-0.006733002341&{}:{}&-0.009725447825&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&4.132129528091&{}:{}&5.820817782772&{}:{}&-8.952947310863&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.624902704110&{}:{}&-0.937354056165&{}:{}&1.312451352055&. \end{alignedat} \]
7a (233)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7a}}&{}\approx{}&1.000694018324&{}:{}&-0.000401806075&{}:{}&-0.000292212249&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.085625991078&{}:{}&-0.049573681585&{}:{}&-0.036052309493&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.013665968550&{}:{}&-0.013369968368&{}:{}&-0.000296000183&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.019839282796&{}:{}&-0.000409493424&{}:{}&-0.019429789372&. \end{alignedat} \]
7a (233)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7a}}&{}\approx{}&0.984741086278&{}:{}&0.006125760478&{}:{}&0.009133153245&,\\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.221782513776&{}:{}&0.312418957663&{}:{}&0.465798528560&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.997356314757&{}:{}&-0.006606470142&{}:{}&0.009250155385&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.003358534485&{}:{}&0.006241573690&{}:{}&-0.009600108175&. \end{alignedat} \]
7a (233)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7a}}&{}\approx{}&1.017025724421&{}:{}&-0.007084358108&{}:{}&-0.009941366313&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.016458450166&{}:{}&-0.006733002341&{}:{}&-0.009725447825&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&4.132129528091&{}:{}&5.820817782772&{}:{}&-8.952947310863&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.624902704110&{}:{}&-0.937354056165&{}:{}&1.312451352055&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.085625991078&{}:{}&-0.049573681585&{}:{}&-0.036052309493&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.013665968550&{}:{}&-0.013369968368&{}:{}&-0.000296000183&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.019839282796&{}:{}&-0.000409493424&{}:{}&-0.019429789372&, \end{alignedat} \]
7a (233)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7a}}&{}\approx{}&0.906149988922&{}:{}&0.067939399643&{}:{}&0.025910611434&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.723914668343&{}:{}&0.276085331657&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.972200722330&{}:{}&0.000000000000&{}:{}&0.027799277670&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.930253423925&{}:{}&0.069746576075&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7a}}&{}\approx{}&1.052293625377&{}:{}&-0.025824990271&{}:{}&-0.026468635106&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.085625991078&{}:{}&-0.049573681585&{}:{}&-0.036052309493&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.013665968550&{}:{}&-0.013369968368&{}:{}&-0.000296000183&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.019839282796&{}:{}&-0.000409493424&{}:{}&-0.019429789372&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.723914668343&{}:{}&0.276085331657&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.972200722330&{}:{}&0.000000000000&{}:{}&0.027799277670&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.930253423925&{}:{}&0.069746576075&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)

Hiroyasu Kamo