Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 108° Top Angle

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

*
[Top] > Isosceles Triangle with 108° Top Angle > 7a (233)

7a(233)

Malfatti circles

7a (233)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.618033988750&{}:{}&0.618033988750&{}:{}&1.000000000000&, \\ P_{\mathbf{7a}}&{}\approx{}&1.002589042964&{}:{}&-0.001401593714&{}:{}&-0.001187449250&, \\ P^-_{\mathbf{7a}}&{}\approx{}&0.975868257024&{}:{}&0.008811642356&{}:{}&0.015320100621&, \\ P^+_{\mathbf{7a}}&{}\approx{}&1.030221019369&{}:{}&-0.011963105592&{}:{}&-0.018257913777&, \\ Q_{\mathbf{7a}}&{}\approx{}&0.857239414698&{}:{}&0.098645138316&{}:{}&0.044115446985&, \\ I^\prime_{\mathbf{7a}}&{}\approx{}&1.088335723635&{}:{}&-0.040692322547&{}:{}&-0.047643401087&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{7a}}\)
\(P^-_{\mathbf{7a}}\)
\(P^+_{\mathbf{7a}}\)
\(Q_{\mathbf{7a}}\)
\(I^\prime_{\mathbf{7a}}\)
7a (233)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7a}}&{}\approx{}&1.028106167787&{}:{}&-0.010735600801&{}:{}&-0.017370566986&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&4.053678004005&{}:{}&3.505310785923&{}:{}&-6.558988789928&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.665147500340&{}:{}&-0.665147500340&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.142781732050&{}:{}&-0.077295734720&{}:{}&-0.065485997329&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.022341000179&{}:{}&-0.021130157050&{}:{}&-0.001210843129&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.036097808054&{}:{}&-0.001448438106&{}:{}&-0.034649369948&, \\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.296898937875&{}:{}&0.256735500016&{}:{}&0.446365562109&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.994770659288&{}:{}&-0.010387507654&{}:{}&0.015616848366&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.007928519814&{}:{}&0.009101131810&{}:{}&-0.017029651624&, \\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.690983005625&{}:{}&0.309016994375&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.951056516295&{}:{}&0.000000000000&{}:{}&0.048943483705&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.896802246667&{}:{}&0.103197753333&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7a}}}{B^\prime_{\mathbf{7a}}}{C^\prime_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^*_{\mathbf{7a}}}{B^*_{\mathbf{7a}}}{C^*_{\mathbf{7a}}}\)
7a (233)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7a}}}}&{}\approx{}&-0.017370566986&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7a}}}}&{}\approx{}&-6.558988789928&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7a}}}}&{}\approx{}&-1.076231263082&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.618033988750&{}:{}&0.618033988750&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.028106167787&{}:{}&-0.010735600801&{}:{}&-0.017370566986&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&4.053678004005&{}:{}&3.505310785923&{}:{}&-6.558988789928&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.665147500340&{}:{}&-0.665147500340&{}:{}&1.000000000000&. \end{alignedat} \]
7a (233)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7a}}&{}\approx{}&1.002589042964&{}:{}&-0.001401593714&{}:{}&-0.001187449250&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.142781732050&{}:{}&-0.077295734720&{}:{}&-0.065485997329&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.022341000179&{}:{}&-0.021130157050&{}:{}&-0.001210843129&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.036097808054&{}:{}&-0.001448438106&{}:{}&-0.034649369948&. \end{alignedat} \]
7a (233)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7a}}&{}\approx{}&0.975868257024&{}:{}&0.008811642356&{}:{}&0.015320100621&,\\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.296898937875&{}:{}&0.256735500016&{}:{}&0.446365562109&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.994770659288&{}:{}&-0.010387507654&{}:{}&0.015616848366&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.007928519814&{}:{}&0.009101131810&{}:{}&-0.017029651624&. \end{alignedat} \]
7a (233)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7a}}&{}\approx{}&1.030221019369&{}:{}&-0.011963105592&{}:{}&-0.018257913777&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.028106167787&{}:{}&-0.010735600801&{}:{}&-0.017370566986&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&4.053678004005&{}:{}&3.505310785923&{}:{}&-6.558988789928&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.665147500340&{}:{}&-0.665147500340&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.142781732050&{}:{}&-0.077295734720&{}:{}&-0.065485997329&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.022341000179&{}:{}&-0.021130157050&{}:{}&-0.001210843129&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.036097808054&{}:{}&-0.001448438106&{}:{}&-0.034649369948&, \end{alignedat} \]
7a (233)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7a}}&{}\approx{}&0.857239414698&{}:{}&0.098645138316&{}:{}&0.044115446985&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.690983005625&{}:{}&0.309016994375&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.951056516295&{}:{}&0.000000000000&{}:{}&0.048943483705&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.896802246667&{}:{}&0.103197753333&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7a}}&{}\approx{}&1.088335723635&{}:{}&-0.040692322547&{}:{}&-0.047643401087&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.142781732050&{}:{}&-0.077295734720&{}:{}&-0.065485997329&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.022341000179&{}:{}&-0.021130157050&{}:{}&-0.001210843129&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.036097808054&{}:{}&-0.001448438106&{}:{}&-0.034649369948&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.690983005625&{}:{}&0.309016994375&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.951056516295&{}:{}&0.000000000000&{}:{}&0.048943483705&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.896802246667&{}:{}&0.103197753333&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)