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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7b (323)

Malfatti circles

7b (323)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.618033988750&{}:{}&-0.618033988750&{}:{}&1.000000000000&, \\ P_{\mathbf{7b}}&{}\approx{}&-0.001401593714&{}:{}&1.002589042964&{}:{}&-0.001187449250&, \\ P^-_{\mathbf{7b}}&{}\approx{}&0.008811642356&{}:{}&0.975868257024&{}:{}&0.015320100621&, \\ P^+_{\mathbf{7b}}&{}\approx{}&-0.011963105592&{}:{}&1.030221019369&{}:{}&-0.018257913777&, \\ Q_{\mathbf{7b}}&{}\approx{}&0.098645138316&{}:{}&0.857239414698&{}:{}&0.044115446985&, \\ I^\prime_{\mathbf{7b}}&{}\approx{}&-0.040692322547&{}:{}&1.088335723635&{}:{}&-0.047643401087&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{7b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{7b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{7b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{7b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{7b}}\) Radical center of the Malfatti circles
7b (323)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7b}}&{}\approx{}&3.505310785923&{}:{}&4.053678004005&{}:{}&-6.558988789928&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.010735600801&{}:{}&1.028106167787&{}:{}&-0.017370566986&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.665147500340&{}:{}&0.665147500340&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.021130157050&{}:{}&1.022341000179&{}:{}&-0.001210843129&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.077295734720&{}:{}&1.142781732050&{}:{}&-0.065485997329&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.001448438106&{}:{}&1.036097808054&{}:{}&-0.034649369948&, \\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.010387507654&{}:{}&0.994770659288&{}:{}&0.015616848366&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.256735500016&{}:{}&0.296898937875&{}:{}&0.446365562109&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.009101131810&{}:{}&1.007928519814&{}:{}&-0.017029651624&, \\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.951056516295&{}:{}&0.048943483705&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.690983005625&{}:{}&0.000000000000&{}:{}&0.309016994375&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.103197753333&{}:{}&0.896802246667&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7b}}}{B^\prime_{\mathbf{7b}}}{C^\prime_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^*_{\mathbf{7b}}}{B^*_{\mathbf{7b}}}{C^*_{\mathbf{7b}}}\)
7b (323)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7b}}}}&{}\approx{}&-6.558988789928&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7b}}}}&{}\approx{}&-0.017370566986&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7b}}}}&{}\approx{}&-1.076231263082&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.618033988750&{}:{}&-0.618033988750&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&3.505310785923&{}:{}&4.053678004005&{}:{}&-6.558988789928&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.010735600801&{}:{}&1.028106167787&{}:{}&-0.017370566986&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.665147500340&{}:{}&0.665147500340&{}:{}&1.000000000000&. \end{alignedat} \]
7b (323)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7b}}&{}\approx{}&-0.001401593714&{}:{}&1.002589042964&{}:{}&-0.001187449250&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.021130157050&{}:{}&1.022341000179&{}:{}&-0.001210843129&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.077295734720&{}:{}&1.142781732050&{}:{}&-0.065485997329&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.001448438106&{}:{}&1.036097808054&{}:{}&-0.034649369948&. \end{alignedat} \]
7b (323)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7b}}&{}\approx{}&0.008811642356&{}:{}&0.975868257024&{}:{}&0.015320100621&,\\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.010387507654&{}:{}&0.994770659288&{}:{}&0.015616848366&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.256735500016&{}:{}&0.296898937875&{}:{}&0.446365562109&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.009101131810&{}:{}&1.007928519814&{}:{}&-0.017029651624&. \end{alignedat} \]
7b (323)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7b}}&{}\approx{}&-0.011963105592&{}:{}&1.030221019369&{}:{}&-0.018257913777&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&3.505310785923&{}:{}&4.053678004005&{}:{}&-6.558988789928&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.010735600801&{}:{}&1.028106167787&{}:{}&-0.017370566986&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.665147500340&{}:{}&0.665147500340&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.021130157050&{}:{}&1.022341000179&{}:{}&-0.001210843129&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.077295734720&{}:{}&1.142781732050&{}:{}&-0.065485997329&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.001448438106&{}:{}&1.036097808054&{}:{}&-0.034649369948&, \end{alignedat} \]
7b (323)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7b}}&{}\approx{}&0.098645138316&{}:{}&0.857239414698&{}:{}&0.044115446985&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.951056516295&{}:{}&0.048943483705&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.690983005625&{}:{}&0.000000000000&{}:{}&0.309016994375&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.103197753333&{}:{}&0.896802246667&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7b}}&{}\approx{}&-0.040692322547&{}:{}&1.088335723635&{}:{}&-0.047643401087&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.021130157050&{}:{}&1.022341000179&{}:{}&-0.001210843129&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.077295734720&{}:{}&1.142781732050&{}:{}&-0.065485997329&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.001448438106&{}:{}&1.036097808054&{}:{}&-0.034649369948&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.951056516295&{}:{}&0.048943483705&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.690983005625&{}:{}&0.000000000000&{}:{}&0.309016994375&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.103197753333&{}:{}&0.896802246667&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)