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 > 6a (231)

6a(231)

Malfatti circles

6a (231)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.618033988750&{}:{}&0.618033988750&{}:{}&1.000000000000&, \\ P_{\mathbf{6a}}&{}\approx{}&1.006240207724&{}:{}&-0.005048434097&{}:{}&-0.001191773627&, \\ P^-_{\mathbf{6a}}&{}\approx{}&0.956066451279&{}:{}&0.014198553756&{}:{}&0.029734994965&, \\ P^+_{\mathbf{6a}}&{}\approx{}&1.059717805309&{}:{}&-0.025562797462&{}:{}&-0.034155007846&, \\ Q_{\mathbf{6a}}&{}\approx{}&1.199810798344&{}:{}&-0.261555730692&{}:{}&0.061744932348&, \\ I^\prime_{\mathbf{6a}}&{}\approx{}&1.129443303240&{}:{}&-0.080000361019&{}:{}&-0.049442942222&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{6a}}\)
\(P^-_{\mathbf{6a}}\)
\(P^+_{\mathbf{6a}}\)
\(Q_{\mathbf{6a}}\)
\(I^\prime_{\mathbf{6a}}\)
6a (231)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6a}}&{}\approx{}&1.055161460168&{}:{}&-0.021069802915&{}:{}&-0.034091657253&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&7.955791036595&{}:{}&5.916949268008&{}:{}&-12.872740304603&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.642039521920&{}:{}&-0.642039521920&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.188190960236&{}:{}&-0.152249685018&{}:{}&-0.035941275217&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.042034819518&{}:{}&-0.040800651367&{}:{}&-0.001234168150&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.073388543238&{}:{}&-0.005385325770&{}:{}&-0.068003217469&, \\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.302914176007&{}:{}&0.225285933699&{}:{}&0.471799890295&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.988989512303&{}:{}&-0.019748460208&{}:{}&0.030758947905&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.017768622871&{}:{}&0.015114893409&{}:{}&-0.032883516280&, \\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.309016994375&{}:{}&-0.309016994375&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.951056516295&{}:{}&0.000000000000&{}:{}&0.048943483705&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.278768257918&{}:{}&-0.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6a}}}{B^\prime_{\mathbf{6a}}}{C^\prime_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^*_{\mathbf{6a}}}{B^*_{\mathbf{6a}}}{C^*_{\mathbf{6a}}}\)
6a (231)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6a}}}}&{}\approx{}&-0.034091657253&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6a}}}}&{}\approx{}&-12.872740304603&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6a}}}}&{}\approx{}&-1.038841768588&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.618033988750&{}:{}&0.618033988750&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.055161460168&{}:{}&-0.021069802915&{}:{}&-0.034091657253&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&7.955791036595&{}:{}&5.916949268008&{}:{}&-12.872740304603&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.642039521920&{}:{}&-0.642039521920&{}:{}&1.000000000000&. \end{alignedat} \]
6a (231)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6a}}&{}\approx{}&1.006240207724&{}:{}&-0.005048434097&{}:{}&-0.001191773627&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.188190960236&{}:{}&-0.152249685018&{}:{}&-0.035941275217&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.042034819518&{}:{}&-0.040800651367&{}:{}&-0.001234168150&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.073388543238&{}:{}&-0.005385325770&{}:{}&-0.068003217469&. \end{alignedat} \]
6a (231)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6a}}&{}\approx{}&0.956066451279&{}:{}&0.014198553756&{}:{}&0.029734994965&,\\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.302914176007&{}:{}&0.225285933699&{}:{}&0.471799890295&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.988989512303&{}:{}&-0.019748460208&{}:{}&0.030758947905&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.017768622871&{}:{}&0.015114893409&{}:{}&-0.032883516280&. \end{alignedat} \]
6a (231)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6a}}&{}\approx{}&1.059717805309&{}:{}&-0.025562797462&{}:{}&-0.034155007846&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.055161460168&{}:{}&-0.021069802915&{}:{}&-0.034091657253&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&7.955791036595&{}:{}&5.916949268008&{}:{}&-12.872740304603&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.642039521920&{}:{}&-0.642039521920&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.188190960236&{}:{}&-0.152249685018&{}:{}&-0.035941275217&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.042034819518&{}:{}&-0.040800651367&{}:{}&-0.001234168150&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.073388543238&{}:{}&-0.005385325770&{}:{}&-0.068003217469&, \end{alignedat} \]
6a (231)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6a}}&{}\approx{}&1.199810798344&{}:{}&-0.261555730692&{}:{}&0.061744932348&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.309016994375&{}:{}&-0.309016994375&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.951056516295&{}:{}&0.000000000000&{}:{}&0.048943483705&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.278768257918&{}:{}&-0.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6a}}&{}\approx{}&1.129443303240&{}:{}&-0.080000361019&{}:{}&-0.049442942222&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.188190960236&{}:{}&-0.152249685018&{}:{}&-0.035941275217&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.042034819518&{}:{}&-0.040800651367&{}:{}&-0.001234168150&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.073388543238&{}:{}&-0.005385325770&{}:{}&-0.068003217469&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.309016994375&{}:{}&-0.309016994375&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.951056516295&{}:{}&0.000000000000&{}:{}&0.048943483705&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.278768257918&{}:{}&-0.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)