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 > 6b (321)

6b(321)

Malfatti circles

6b (321)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.618033988750&{}:{}&-0.618033988750&{}:{}&1.000000000000&, \\ P_{\mathbf{6b}}&{}\approx{}&-0.005048434097&{}:{}&1.006240207724&{}:{}&-0.001191773627&, \\ P^-_{\mathbf{6b}}&{}\approx{}&0.014198553756&{}:{}&0.956066451279&{}:{}&0.029734994965&, \\ P^+_{\mathbf{6b}}&{}\approx{}&-0.025562797462&{}:{}&1.059717805309&{}:{}&-0.034155007846&, \\ Q_{\mathbf{6b}}&{}\approx{}&-0.261555730692&{}:{}&1.199810798344&{}:{}&0.061744932348&, \\ I^\prime_{\mathbf{6b}}&{}\approx{}&-0.080000361019&{}:{}&1.129443303240&{}:{}&-0.049442942222&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{6b}}\)
\(P^-_{\mathbf{6b}}\)
\(P^+_{\mathbf{6b}}\)
\(Q_{\mathbf{6b}}\)
\(I^\prime_{\mathbf{6b}}\)
6b (321)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6b}}&{}\approx{}&5.916949268008&{}:{}&7.955791036595&{}:{}&-12.872740304603&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.021069802915&{}:{}&1.055161460168&{}:{}&-0.034091657253&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.642039521920&{}:{}&0.642039521920&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.040800651367&{}:{}&1.042034819518&{}:{}&-0.001234168150&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.152249685018&{}:{}&1.188190960236&{}:{}&-0.035941275217&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.005385325770&{}:{}&1.073388543238&{}:{}&-0.068003217469&, \\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.019748460208&{}:{}&0.988989512303&{}:{}&0.030758947905&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.225285933699&{}:{}&0.302914176007&{}:{}&0.471799890295&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.015114893409&{}:{}&1.017768622871&{}:{}&-0.032883516280&, \\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.951056516295&{}:{}&0.048943483705&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.309016994375&{}:{}&0.000000000000&{}:{}&-0.309016994375&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.278768257918&{}:{}&1.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6b}}}{B^\prime_{\mathbf{6b}}}{C^\prime_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^*_{\mathbf{6b}}}{B^*_{\mathbf{6b}}}{C^*_{\mathbf{6b}}}\)
6b (321)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6b}}}}&{}\approx{}&-12.872740304603&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6b}}}}&{}\approx{}&-0.034091657253&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6b}}}}&{}\approx{}&-1.038841768588&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.618033988750&{}:{}&-0.618033988750&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&5.916949268008&{}:{}&7.955791036595&{}:{}&-12.872740304603&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.021069802915&{}:{}&1.055161460168&{}:{}&-0.034091657253&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.642039521920&{}:{}&0.642039521920&{}:{}&1.000000000000&. \end{alignedat} \]
6b (321)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6b}}&{}\approx{}&-0.005048434097&{}:{}&1.006240207724&{}:{}&-0.001191773627&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.040800651367&{}:{}&1.042034819518&{}:{}&-0.001234168150&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.152249685018&{}:{}&1.188190960236&{}:{}&-0.035941275217&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.005385325770&{}:{}&1.073388543238&{}:{}&-0.068003217469&. \end{alignedat} \]
6b (321)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6b}}&{}\approx{}&0.014198553756&{}:{}&0.956066451279&{}:{}&0.029734994965&,\\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.019748460208&{}:{}&0.988989512303&{}:{}&0.030758947905&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.225285933699&{}:{}&0.302914176007&{}:{}&0.471799890295&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.015114893409&{}:{}&1.017768622871&{}:{}&-0.032883516280&. \end{alignedat} \]
6b (321)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6b}}&{}\approx{}&-0.025562797462&{}:{}&1.059717805309&{}:{}&-0.034155007846&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&5.916949268008&{}:{}&7.955791036595&{}:{}&-12.872740304603&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.021069802915&{}:{}&1.055161460168&{}:{}&-0.034091657253&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.642039521920&{}:{}&0.642039521920&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.040800651367&{}:{}&1.042034819518&{}:{}&-0.001234168150&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.152249685018&{}:{}&1.188190960236&{}:{}&-0.035941275217&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.005385325770&{}:{}&1.073388543238&{}:{}&-0.068003217469&, \end{alignedat} \]
6b (321)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6b}}&{}\approx{}&-0.261555730692&{}:{}&1.199810798344&{}:{}&0.061744932348&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.951056516295&{}:{}&0.048943483705&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.309016994375&{}:{}&0.000000000000&{}:{}&-0.309016994375&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.278768257918&{}:{}&1.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6b}}&{}\approx{}&-0.080000361019&{}:{}&1.129443303240&{}:{}&-0.049442942222&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.040800651367&{}:{}&1.042034819518&{}:{}&-0.001234168150&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.152249685018&{}:{}&1.188190960236&{}:{}&-0.035941275217&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.005385325770&{}:{}&1.073388543238&{}:{}&-0.068003217469&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.951056516295&{}:{}&0.048943483705&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.309016994375&{}:{}&0.000000000000&{}:{}&-0.309016994375&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.278768257918&{}:{}&1.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)