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 > 2a (031)

2a(031)

Malfatti circles

2a (031)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.707106781187&{}:{}&0.707106781187&{}:{}&1.000000000000&, \\ P_{\mathbf{2a}}&{}\approx{}&-0.016141216526&{}:{}&0.031249746162&{}:{}&0.984891470364&, \\ P^-_{\mathbf{2a}}&{}\approx{}&0.023471422383&{}:{}&-0.007496729859&{}:{}&0.984025307476&, \\ P^+_{\mathbf{2a}}&{}\approx{}&-0.051679122071&{}:{}&0.066010586211&{}:{}&0.985668535859&, \\ Q_{\mathbf{2a}}&{}\approx{}&0.102271340830&{}:{}&-0.343545884624&{}:{}&1.241274543794&, \\ I^\prime_{\mathbf{2a}}&{}\approx{}&-0.103298079348&{}:{}&0.143729978934&{}:{}&0.959568100414&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{2a}}\)
\(P^-_{\mathbf{2a}}\)
\(P^+_{\mathbf{2a}}\)
\(Q_{\mathbf{2a}}\)
\(I^\prime_{\mathbf{2a}}\)
2a (031)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2a}}&{}\approx{}&0.193437035124&{}:{}&0.334089318960&{}:{}&0.472473645917&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-4.361428811086&{}:{}&-0.806562964876&{}:{}&6.167991775962&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.041196100146&{}:{}&0.041196100146&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.081621242647&{}:{}&0.033263476303&{}:{}&1.048357766344&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.015439783364&{}:{}&0.073347784598&{}:{}&0.942091998766&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.130644777866&{}:{}&0.252931130636&{}:{}&0.877713647231&, \\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.043310270748&{}:{}&-0.008009407444&{}:{}&1.051319678192&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.022395653121&{}:{}&0.038680020576&{}:{}&0.938924326303&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.320192538453&{}:{}&-0.102268917680&{}:{}&0.782076379228&, \\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.382683432365&{}:{}&1.382683432365&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.076120467489&{}:{}&0.000000000000&{}:{}&0.923879532511&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.423879532511&{}:{}&1.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2a}}}{B^\prime_{\mathbf{2a}}}{C^\prime_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^*_{\mathbf{2a}}}{B^*_{\mathbf{2a}}}{C^*_{\mathbf{2a}}}\)
2a (031)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2a}}}}&{}\approx{}&0.472473645917&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2a}}}}&{}\approx{}&6.167991775962&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2a}}}}&{}\approx{}&0.058260083544&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.707106781187&{}:{}&0.707106781187&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.193437035124&{}:{}&0.334089318960&{}:{}&0.472473645917&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-4.361428811086&{}:{}&-0.806562964876&{}:{}&6.167991775962&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.041196100146&{}:{}&0.041196100146&{}:{}&1.000000000000&. \end{alignedat} \]
2a (031)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2a}}&{}\approx{}&-0.016141216526&{}:{}&0.031249746162&{}:{}&0.984891470364&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.081621242647&{}:{}&0.033263476303&{}:{}&1.048357766344&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.015439783364&{}:{}&0.073347784598&{}:{}&0.942091998766&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.130644777866&{}:{}&0.252931130636&{}:{}&0.877713647231&. \end{alignedat} \]
2a (031)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2a}}&{}\approx{}&0.023471422383&{}:{}&-0.007496729859&{}:{}&0.984025307476&,\\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.043310270748&{}:{}&-0.008009407444&{}:{}&1.051319678192&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.022395653121&{}:{}&0.038680020576&{}:{}&0.938924326303&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.320192538453&{}:{}&-0.102268917680&{}:{}&0.782076379228&. \end{alignedat} \]
2a (031)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2a}}&{}\approx{}&-0.051679122071&{}:{}&0.066010586211&{}:{}&0.985668535859&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.193437035124&{}:{}&0.334089318960&{}:{}&0.472473645917&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-4.361428811086&{}:{}&-0.806562964876&{}:{}&6.167991775962&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.041196100146&{}:{}&0.041196100146&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.081621242647&{}:{}&0.033263476303&{}:{}&1.048357766344&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.015439783364&{}:{}&0.073347784598&{}:{}&0.942091998766&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.130644777866&{}:{}&0.252931130636&{}:{}&0.877713647231&, \end{alignedat} \]
2a (031)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2a}}&{}\approx{}&0.102271340830&{}:{}&-0.343545884624&{}:{}&1.241274543794&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.382683432365&{}:{}&1.382683432365&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.076120467489&{}:{}&0.000000000000&{}:{}&0.923879532511&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.423879532511&{}:{}&1.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2a}}&{}\approx{}&-0.103298079348&{}:{}&0.143729978934&{}:{}&0.959568100414&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.081621242647&{}:{}&0.033263476303&{}:{}&1.048357766344&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.015439783364&{}:{}&0.073347784598&{}:{}&0.942091998766&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.130644777866&{}:{}&0.252931130636&{}:{}&0.877713647231&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.382683432365&{}:{}&1.382683432365&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.076120467489&{}:{}&0.000000000000&{}:{}&0.923879532511&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.423879532511&{}:{}&1.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)