Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 90° Top Angle

\(A=45\degree\), \(B=45\degree\), \(C=90\degree\).


[Top] > Isosceles Triangle with 90° Top Angle > 7b (323)

7b(323)

Malfatti circles

7b (323)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.707106781187&{}:{}&-0.707106781187&{}:{}&1.000000000000&, \\ P_{\mathbf{7b}}&{}\approx{}&-0.003048612742&{}:{}&1.005877016473&{}:{}&-0.002828403731&, \\ P^-_{\mathbf{7b}}&{}\approx{}&0.012875386094&{}:{}&0.967466335890&{}:{}&0.019658278016&, \\ P^+_{\mathbf{7b}}&{}\approx{}&-0.019720277726&{}:{}&1.046091161492&{}:{}&-0.026370883766&, \\ Q_{\mathbf{7b}}&{}\approx{}&0.109363008321&{}:{}&0.822841287509&{}:{}&0.067795704169&, \\ I^\prime_{\mathbf{7b}}&{}\approx{}&-0.062426032111&{}:{}&1.133932084180&{}:{}&-0.071506052069&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{7b}}\)
\(P^-_{\mathbf{7b}}\)
\(P^+_{\mathbf{7b}}\)
\(Q_{\mathbf{7b}}\)
\(I^\prime_{\mathbf{7b}}\)
7b (323)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7b}}&{}\approx{}&2.041196100146&{}:{}&2.513669746063&{}:{}&-3.554865846209&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.017063983397&{}:{}&1.041196100146&{}:{}&-0.024132116749&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.806562964876&{}:{}&0.806562964876&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.033420900252&{}:{}&1.036334947979&{}:{}&-0.002914047727&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.114775010625&{}:{}&1.221259531854&{}:{}&-0.106484521229&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.003185749001&{}:{}&1.051124551154&{}:{}&-0.047938802153&, \\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.016324663312&{}:{}&0.996084875443&{}:{}&0.020239787869&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.266079180035&{}:{}&0.327668265123&{}:{}&0.406252554843&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.013441004221&{}:{}&1.009967313500&{}:{}&-0.023408317721&, \\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.923879532511&{}:{}&0.076120467489&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.617316567635&{}:{}&0.000000000000&{}:{}&0.382683432365&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.117316567635&{}:{}&0.882683432365&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7b}}}{B^\prime_{\mathbf{7b}}}{C^\prime_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^*_{\mathbf{7b}}}{B^*_{\mathbf{7b}}}{C^*_{\mathbf{7b}}}\)
7b (323)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7b}}}}&{}\approx{}&-3.554865846209&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7b}}}}&{}\approx{}&-0.024132116749&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7b}}}}&{}\approx{}&-1.140652283836&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.707106781187&{}:{}&-0.707106781187&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&2.041196100146&{}:{}&2.513669746063&{}:{}&-3.554865846209&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.017063983397&{}:{}&1.041196100146&{}:{}&-0.024132116749&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.806562964876&{}:{}&0.806562964876&{}:{}&1.000000000000&. \end{alignedat} \]
7b (323)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7b}}&{}\approx{}&-0.003048612742&{}:{}&1.005877016473&{}:{}&-0.002828403731&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.033420900252&{}:{}&1.036334947979&{}:{}&-0.002914047727&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.114775010625&{}:{}&1.221259531854&{}:{}&-0.106484521229&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.003185749001&{}:{}&1.051124551154&{}:{}&-0.047938802153&. \end{alignedat} \]
7b (323)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7b}}&{}\approx{}&0.012875386094&{}:{}&0.967466335890&{}:{}&0.019658278016&,\\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.016324663312&{}:{}&0.996084875443&{}:{}&0.020239787869&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.266079180035&{}:{}&0.327668265123&{}:{}&0.406252554843&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.013441004221&{}:{}&1.009967313500&{}:{}&-0.023408317721&. \end{alignedat} \]
7b (323)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7b}}&{}\approx{}&-0.019720277726&{}:{}&1.046091161492&{}:{}&-0.026370883766&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&2.041196100146&{}:{}&2.513669746063&{}:{}&-3.554865846209&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.017063983397&{}:{}&1.041196100146&{}:{}&-0.024132116749&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.806562964876&{}:{}&0.806562964876&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.033420900252&{}:{}&1.036334947979&{}:{}&-0.002914047727&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.114775010625&{}:{}&1.221259531854&{}:{}&-0.106484521229&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.003185749001&{}:{}&1.051124551154&{}:{}&-0.047938802153&, \end{alignedat} \]
7b (323)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7b}}&{}\approx{}&0.109363008321&{}:{}&0.822841287509&{}:{}&0.067795704169&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.923879532511&{}:{}&0.076120467489&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.617316567635&{}:{}&0.000000000000&{}:{}&0.382683432365&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.117316567635&{}:{}&0.882683432365&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7b}}&{}\approx{}&-0.062426032111&{}:{}&1.133932084180&{}:{}&-0.071506052069&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.033420900252&{}:{}&1.036334947979&{}:{}&-0.002914047727&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.114775010625&{}:{}&1.221259531854&{}:{}&-0.106484521229&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.003185749001&{}:{}&1.051124551154&{}:{}&-0.047938802153&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.923879532511&{}:{}&0.076120467489&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.617316567635&{}:{}&0.000000000000&{}:{}&0.382683432365&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.117316567635&{}:{}&0.882683432365&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)