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 > 7a (233)

7a(233)

Malfatti circles

7a (233)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.707106781187&{}:{}&0.707106781187&{}:{}&1.000000000000&, \\ P_{\mathbf{7a}}&{}\approx{}&1.005877016473&{}:{}&-0.003048612742&{}:{}&-0.002828403731&, \\ P^-_{\mathbf{7a}}&{}\approx{}&0.967466335890&{}:{}&0.012875386094&{}:{}&0.019658278016&, \\ P^+_{\mathbf{7a}}&{}\approx{}&1.046091161492&{}:{}&-0.019720277726&{}:{}&-0.026370883766&, \\ Q_{\mathbf{7a}}&{}\approx{}&0.822841287509&{}:{}&0.109363008321&{}:{}&0.067795704169&, \\ I^\prime_{\mathbf{7a}}&{}\approx{}&1.133932084180&{}:{}&-0.062426032111&{}:{}&-0.071506052069&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{7a}}\)
\(P^-_{\mathbf{7a}}\)
\(P^+_{\mathbf{7a}}\)
\(Q_{\mathbf{7a}}\)
\(I^\prime_{\mathbf{7a}}\)
7a (233)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7a}}&{}\approx{}&1.041196100146&{}:{}&-0.017063983397&{}:{}&-0.024132116749&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&2.513669746063&{}:{}&2.041196100146&{}:{}&-3.554865846209&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.806562964876&{}:{}&-0.806562964876&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.221259531854&{}:{}&-0.114775010625&{}:{}&-0.106484521229&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.036334947979&{}:{}&-0.033420900252&{}:{}&-0.002914047727&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.051124551154&{}:{}&-0.003185749001&{}:{}&-0.047938802153&, \\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.327668265123&{}:{}&0.266079180035&{}:{}&0.406252554843&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.996084875443&{}:{}&-0.016324663312&{}:{}&0.020239787869&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.009967313500&{}:{}&0.013441004221&{}:{}&-0.023408317721&, \\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.617316567635&{}:{}&0.382683432365&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.923879532511&{}:{}&0.000000000000&{}:{}&0.076120467489&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.882683432365&{}:{}&0.117316567635&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7a}}}{B^\prime_{\mathbf{7a}}}{C^\prime_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^*_{\mathbf{7a}}}{B^*_{\mathbf{7a}}}{C^*_{\mathbf{7a}}}\)
7a (233)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7a}}}}&{}\approx{}&-0.024132116749&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7a}}}}&{}\approx{}&-3.554865846209&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7a}}}}&{}\approx{}&-1.140652283836&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.707106781187&{}:{}&0.707106781187&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.041196100146&{}:{}&-0.017063983397&{}:{}&-0.024132116749&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&2.513669746063&{}:{}&2.041196100146&{}:{}&-3.554865846209&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.806562964876&{}:{}&-0.806562964876&{}:{}&1.000000000000&. \end{alignedat} \]
7a (233)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7a}}&{}\approx{}&1.005877016473&{}:{}&-0.003048612742&{}:{}&-0.002828403731&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.221259531854&{}:{}&-0.114775010625&{}:{}&-0.106484521229&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.036334947979&{}:{}&-0.033420900252&{}:{}&-0.002914047727&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.051124551154&{}:{}&-0.003185749001&{}:{}&-0.047938802153&. \end{alignedat} \]
7a (233)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7a}}&{}\approx{}&0.967466335890&{}:{}&0.012875386094&{}:{}&0.019658278016&,\\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.327668265123&{}:{}&0.266079180035&{}:{}&0.406252554843&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.996084875443&{}:{}&-0.016324663312&{}:{}&0.020239787869&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.009967313500&{}:{}&0.013441004221&{}:{}&-0.023408317721&. \end{alignedat} \]
7a (233)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7a}}&{}\approx{}&1.046091161492&{}:{}&-0.019720277726&{}:{}&-0.026370883766&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.041196100146&{}:{}&-0.017063983397&{}:{}&-0.024132116749&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&2.513669746063&{}:{}&2.041196100146&{}:{}&-3.554865846209&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.806562964876&{}:{}&-0.806562964876&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.221259531854&{}:{}&-0.114775010625&{}:{}&-0.106484521229&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.036334947979&{}:{}&-0.033420900252&{}:{}&-0.002914047727&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.051124551154&{}:{}&-0.003185749001&{}:{}&-0.047938802153&, \end{alignedat} \]
7a (233)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7a}}&{}\approx{}&0.822841287509&{}:{}&0.109363008321&{}:{}&0.067795704169&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.617316567635&{}:{}&0.382683432365&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.923879532511&{}:{}&0.000000000000&{}:{}&0.076120467489&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.882683432365&{}:{}&0.117316567635&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7a}}&{}\approx{}&1.133932084180&{}:{}&-0.062426032111&{}:{}&-0.071506052069&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.221259531854&{}:{}&-0.114775010625&{}:{}&-0.106484521229&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.036334947979&{}:{}&-0.033420900252&{}:{}&-0.002914047727&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.051124551154&{}:{}&-0.003185749001&{}:{}&-0.047938802153&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.617316567635&{}:{}&0.382683432365&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.923879532511&{}:{}&0.000000000000&{}:{}&0.076120467489&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.882683432365&{}:{}&0.117316567635&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)