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 > 4b (301)

4b(301)

Malfatti circles

4b (301)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.707106781187&{}:{}&-0.707106781187&{}:{}&1.000000000000&, \\ P_{\mathbf{4b}}&{}\approx{}&0.031249746162&{}:{}&-0.016141216526&{}:{}&0.984891470364&, \\ P^-_{\mathbf{4b}}&{}\approx{}&-0.007496729859&{}:{}&0.023471422383&{}:{}&0.984025307476&, \\ P^+_{\mathbf{4b}}&{}\approx{}&0.066010586211&{}:{}&-0.051679122071&{}:{}&0.985668535859&, \\ Q_{\mathbf{4b}}&{}\approx{}&-0.343545884624&{}:{}&0.102271340830&{}:{}&1.241274543794&, \\ I^\prime_{\mathbf{4b}}&{}\approx{}&0.143729978934&{}:{}&-0.103298079348&{}:{}&0.959568100414&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{4b}}\)
\(P^-_{\mathbf{4b}}\)
\(P^+_{\mathbf{4b}}\)
\(Q_{\mathbf{4b}}\)
\(I^\prime_{\mathbf{4b}}\)
4b (301)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4b}}&{}\approx{}&-0.806562964876&{}:{}&-4.361428811086&{}:{}&6.167991775962&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.334089318960&{}:{}&0.193437035124&{}:{}&0.472473645917&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.041196100146&{}:{}&-0.041196100146&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.073347784598&{}:{}&-0.015439783364&{}:{}&0.942091998766&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.033263476303&{}:{}&-0.081621242647&{}:{}&1.048357766344&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.252931130636&{}:{}&-0.130644777866&{}:{}&0.877713647231&, \\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.038680020576&{}:{}&0.022395653121&{}:{}&0.938924326303&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.008009407444&{}:{}&-0.043310270748&{}:{}&1.051319678192&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.102268917680&{}:{}&0.320192538453&{}:{}&0.782076379228&, \\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.076120467489&{}:{}&0.923879532511&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.382683432365&{}:{}&0.000000000000&{}:{}&1.382683432365&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.423879532511&{}:{}&-0.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4b}}}{B^\prime_{\mathbf{4b}}}{C^\prime_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^*_{\mathbf{4b}}}{B^*_{\mathbf{4b}}}{C^*_{\mathbf{4b}}}\)
4b (301)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4b}}}}&{}\approx{}&6.167991775962&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4b}}}}&{}\approx{}&0.472473645917&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4b}}}}&{}\approx{}&0.058260083544&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.707106781187&{}:{}&-0.707106781187&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&-0.806562964876&{}:{}&-4.361428811086&{}:{}&6.167991775962&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.334089318960&{}:{}&0.193437035124&{}:{}&0.472473645917&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.041196100146&{}:{}&-0.041196100146&{}:{}&1.000000000000&. \end{alignedat} \]
4b (301)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4b}}&{}\approx{}&0.031249746162&{}:{}&-0.016141216526&{}:{}&0.984891470364&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.073347784598&{}:{}&-0.015439783364&{}:{}&0.942091998766&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.033263476303&{}:{}&-0.081621242647&{}:{}&1.048357766344&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.252931130636&{}:{}&-0.130644777866&{}:{}&0.877713647231&. \end{alignedat} \]
4b (301)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4b}}&{}\approx{}&-0.007496729859&{}:{}&0.023471422383&{}:{}&0.984025307476&,\\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.038680020576&{}:{}&0.022395653121&{}:{}&0.938924326303&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.008009407444&{}:{}&-0.043310270748&{}:{}&1.051319678192&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.102268917680&{}:{}&0.320192538453&{}:{}&0.782076379228&. \end{alignedat} \]
4b (301)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4b}}&{}\approx{}&0.066010586211&{}:{}&-0.051679122071&{}:{}&0.985668535859&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&-0.806562964876&{}:{}&-4.361428811086&{}:{}&6.167991775962&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.334089318960&{}:{}&0.193437035124&{}:{}&0.472473645917&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.041196100146&{}:{}&-0.041196100146&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.073347784598&{}:{}&-0.015439783364&{}:{}&0.942091998766&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.033263476303&{}:{}&-0.081621242647&{}:{}&1.048357766344&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.252931130636&{}:{}&-0.130644777866&{}:{}&0.877713647231&, \end{alignedat} \]
4b (301)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4b}}&{}\approx{}&-0.343545884624&{}:{}&0.102271340830&{}:{}&1.241274543794&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.076120467489&{}:{}&0.923879532511&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.382683432365&{}:{}&0.000000000000&{}:{}&1.382683432365&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.423879532511&{}:{}&-0.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4b}}&{}\approx{}&0.143729978934&{}:{}&-0.103298079348&{}:{}&0.959568100414&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.073347784598&{}:{}&-0.015439783364&{}:{}&0.942091998766&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.033263476303&{}:{}&-0.081621242647&{}:{}&1.048357766344&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.252931130636&{}:{}&-0.130644777866&{}:{}&0.877713647231&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.076120467489&{}:{}&0.923879532511&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.382683432365&{}:{}&0.000000000000&{}:{}&1.382683432365&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.423879532511&{}:{}&-0.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)