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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\(\mathbf{3a}\) \((033)\)

Malfatti circles

3a (033)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.375000000000&{}:{}&0.562500000000&{}:{}&0.812500000000&, \\ P_{\mathbf{3a}}&{}\approx{}&-0.002229600710&{}:{}&0.028330603607&{}:{}&0.973898997103&, \\ P^-_{\mathbf{3a}}&{}\approx{}&0.022450749985&{}:{}&-0.007035638076&{}:{}&0.984584888091&, \\ P^+_{\mathbf{3a}}&{}\approx{}&-0.024024024690&{}:{}&0.059561394260&{}:{}&0.964462630430&, \\ Q_{\mathbf{3a}}&{}\approx{}&0.020279952962&{}:{}&0.270486334118&{}:{}&0.709233712920&, \\ I^\prime_{\mathbf{3a}}&{}\approx{}&-0.029300144219&{}:{}&0.127917438528&{}:{}&0.901382705691&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{3a}}\)
\(P^-_{\mathbf{3a}}\)
\(P^+_{\mathbf{3a}}\)
\(Q_{\mathbf{3a}}\)
\(I^\prime_{\mathbf{3a}}\)
3a (033)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3a}}&{}\approx{}&0.197827264120&{}:{}&0.328161573769&{}:{}&0.474011162111&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-1.132129528091&{}:{}&-0.320817782772&{}:{}&2.452947310863&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.024902704110&{}:{}&0.037354056165&{}:{}&0.987548647945&, \\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.048733450750&{}:{}&0.029645154824&{}:{}&1.019088295925&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.002140701444&{}:{}&0.067073278699&{}:{}&0.935067422746&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.017554802437&{}:{}&0.223061531585&{}:{}&0.794493270851&, \\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.026060053842&{}:{}&-0.007384781057&{}:{}&1.033444834899&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.021498833453&{}:{}&0.035662885253&{}:{}&0.942838281294&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.324390415097&{}:{}&-0.101657786824&{}:{}&0.777267371727&, \\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.276085331657&{}:{}&0.723914668343&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.027799277670&{}:{}&0.000000000000&{}:{}&0.972200722330&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.069746576075&{}:{}&0.930253423925&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3a}}}{B^\prime_{\mathbf{3a}}}{C^\prime_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^*_{\mathbf{3a}}}{B^*_{\mathbf{3a}}}{C^*_{\mathbf{3a}}}\)
3a (033)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3a}}}}&{}\approx{}&0.583398353368&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3a}}}}&{}\approx{}&3.019012074908&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3a}}}}&{}\approx{}&0.066407210959&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.375000000000&{}:{}&0.562500000000&{}:{}&0.812500000000&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.197827264120&{}:{}&0.328161573769&{}:{}&0.474011162111&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-1.132129528091&{}:{}&-0.320817782772&{}:{}&2.452947310863&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.024902704110&{}:{}&0.037354056165&{}:{}&0.987548647945&. \end{alignedat} \]
3a (033)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3a}}&{}\approx{}&-0.002229600710&{}:{}&0.028330603607&{}:{}&0.973898997103&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.048733450750&{}:{}&0.029645154824&{}:{}&1.019088295925&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.002140701444&{}:{}&0.067073278699&{}:{}&0.935067422746&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.017554802437&{}:{}&0.223061531585&{}:{}&0.794493270851&. \end{alignedat} \]
3a (033)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3a}}&{}\approx{}&0.022450749985&{}:{}&-0.007035638076&{}:{}&0.984584888091&,\\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.026060053842&{}:{}&-0.007384781057&{}:{}&1.033444834899&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.021498833453&{}:{}&0.035662885253&{}:{}&0.942838281294&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.324390415097&{}:{}&-0.101657786824&{}:{}&0.777267371727&. \end{alignedat} \]
3a (033)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3a}}&{}\approx{}&-0.024024024690&{}:{}&0.059561394260&{}:{}&0.964462630430&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.197827264120&{}:{}&0.328161573769&{}:{}&0.474011162111&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-1.132129528091&{}:{}&-0.320817782772&{}:{}&2.452947310863&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.024902704110&{}:{}&0.037354056165&{}:{}&0.987548647945&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.048733450750&{}:{}&0.029645154824&{}:{}&1.019088295925&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.002140701444&{}:{}&0.067073278699&{}:{}&0.935067422746&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.017554802437&{}:{}&0.223061531585&{}:{}&0.794493270851&, \end{alignedat} \]
3a (033)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3a}}&{}\approx{}&0.020279952962&{}:{}&0.270486334118&{}:{}&0.709233712920&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.276085331657&{}:{}&0.723914668343&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.027799277670&{}:{}&0.000000000000&{}:{}&0.972200722330&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.069746576075&{}:{}&0.930253423925&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3a}}&{}\approx{}&-0.029300144219&{}:{}&0.127917438528&{}:{}&0.901382705691&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.048733450750&{}:{}&0.029645154824&{}:{}&1.019088295925&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.002140701444&{}:{}&0.067073278699&{}:{}&0.935067422746&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.017554802437&{}:{}&0.223061531585&{}:{}&0.794493270851&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.276085331657&{}:{}&0.723914668343&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.027799277670&{}:{}&0.000000000000&{}:{}&0.972200722330&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.069746576075&{}:{}&0.930253423925&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)

Hiroyasu Kamo