Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 108° Top Angle

\(A=36\degree\), \(B=36\degree\), \(C=108\degree\).

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[Top] > Isosceles Triangle with 108° Top Angle > 3a (033)

3a(033)

Malfatti circles

3a (033)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.618033988750&{}:{}&0.618033988750&{}:{}&1.000000000000&, \\ P_{\mathbf{3a}}&{}\approx{}&-0.005687084864&{}:{}&0.045341373936&{}:{}&0.960345710928&, \\ P^-_{\mathbf{3a}}&{}\approx{}&0.039006726956&{}:{}&0.003541838746&{}:{}&0.957451434297&, \\ P^+_{\mathbf{3a}}&{}\approx{}&-0.044687755655&{}:{}&0.081816443856&{}:{}&0.962871311798&, \\ Q_{\mathbf{3a}}&{}\approx{}&0.034338463035&{}:{}&0.298405825736&{}:{}&0.667255711228&, \\ I^\prime_{\mathbf{3a}}&{}\approx{}&-0.054486376717&{}:{}&0.153847488979&{}:{}&0.900638887737&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{3a}}\)
\(P^-_{\mathbf{3a}}\)
\(P^+_{\mathbf{3a}}\)
\(Q_{\mathbf{3a}}\)
\(I^\prime_{\mathbf{3a}}\)
3a (033)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3a}}&{}\approx{}&0.248287034463&{}:{}&0.287128803051&{}:{}&0.464584162486&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-1.435644015255&{}:{}&0.112723202827&{}:{}&2.322920812428&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.047113511590&{}:{}&0.047113511590&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.091233029983&{}:{}&0.049198210466&{}:{}&1.042034819518&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.005474986363&{}:{}&0.080945144273&{}:{}&0.924529842089&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.032368140490&{}:{}&0.258061203014&{}:{}&0.774306937476&, \\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.049251740217&{}:{}&0.003867124331&{}:{}&1.045384615886&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.037450038923&{}:{}&0.043308684537&{}:{}&0.919241276541&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.337614478500&{}:{}&0.030655636465&{}:{}&0.631729885035&, \\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.309016994375&{}:{}&0.690983005625&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.048943483705&{}:{}&0.000000000000&{}:{}&0.951056516295&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.103197753333&{}:{}&0.896802246667&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3a}}}{B^\prime_{\mathbf{3a}}}{C^\prime_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^*_{\mathbf{3a}}}{B^*_{\mathbf{3a}}}{C^*_{\mathbf{3a}}}\)
3a (033)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3a}}}}&{}\approx{}&0.464584162486&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3a}}}}&{}\approx{}&2.322920812428&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3a}}}}&{}\approx{}&0.076231263082&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.618033988750&{}:{}&0.618033988750&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.248287034463&{}:{}&0.287128803051&{}:{}&0.464584162486&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-1.435644015255&{}:{}&0.112723202827&{}:{}&2.322920812428&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.047113511590&{}:{}&0.047113511590&{}:{}&1.000000000000&. \end{alignedat} \]
3a (033)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3a}}&{}\approx{}&-0.005687084864&{}:{}&0.045341373936&{}:{}&0.960345710928&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.091233029983&{}:{}&0.049198210466&{}:{}&1.042034819518&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.005474986363&{}:{}&0.080945144273&{}:{}&0.924529842089&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.032368140490&{}:{}&0.258061203014&{}:{}&0.774306937476&. \end{alignedat} \]
3a (033)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3a}}&{}\approx{}&0.039006726956&{}:{}&0.003541838746&{}:{}&0.957451434297&,\\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.049251740217&{}:{}&0.003867124331&{}:{}&1.045384615886&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.037450038923&{}:{}&0.043308684537&{}:{}&0.919241276541&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.337614478500&{}:{}&0.030655636465&{}:{}&0.631729885035&. \end{alignedat} \]
3a (033)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3a}}&{}\approx{}&-0.044687755655&{}:{}&0.081816443856&{}:{}&0.962871311798&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.248287034463&{}:{}&0.287128803051&{}:{}&0.464584162486&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-1.435644015255&{}:{}&0.112723202827&{}:{}&2.322920812428&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.047113511590&{}:{}&0.047113511590&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.091233029983&{}:{}&0.049198210466&{}:{}&1.042034819518&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.005474986363&{}:{}&0.080945144273&{}:{}&0.924529842089&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.032368140490&{}:{}&0.258061203014&{}:{}&0.774306937476&, \end{alignedat} \]
3a (033)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3a}}&{}\approx{}&0.034338463035&{}:{}&0.298405825736&{}:{}&0.667255711228&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.309016994375&{}:{}&0.690983005625&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.048943483705&{}:{}&0.000000000000&{}:{}&0.951056516295&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.103197753333&{}:{}&0.896802246667&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3a}}&{}\approx{}&-0.054486376717&{}:{}&0.153847488979&{}:{}&0.900638887737&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.091233029983&{}:{}&0.049198210466&{}:{}&1.042034819518&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.005474986363&{}:{}&0.080945144273&{}:{}&0.924529842089&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.032368140490&{}:{}&0.258061203014&{}:{}&0.774306937476&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.309016994375&{}:{}&0.690983005625&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.048943483705&{}:{}&0.000000000000&{}:{}&0.951056516295&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.103197753333&{}:{}&0.896802246667&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)