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 > 3b (123)

3b (123)

Malfatti circles

3b (123)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.707106781187&{}:{}&-0.707106781187&{}:{}&1.000000000000&, \\ P_{\mathbf{3b}}&{}\approx{}&-0.003362146569&{}:{}&1.109326190582&{}:{}&-0.105964044013&, \\ P^-_{\mathbf{3b}}&{}\approx{}&0.087929062579&{}:{}&0.875924903275&{}:{}&0.036146034145&, \\ P^+_{\mathbf{3b}}&{}\approx{}&-0.126228645930&{}:{}&1.423455003042&{}:{}&-0.297226357112&, \\ Q_{\mathbf{3b}}&{}\approx{}&0.203632007983&{}:{}&1.532116080188&{}:{}&-0.735748088171&, \\ I^\prime_{\mathbf{3b}}&{}\approx{}&-0.095345022530&{}:{}&1.731886145206&{}:{}&-0.636541122676&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{3b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{3b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{3b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{3b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{3b}}\) Radical center of the Malfatti circles
3b (123)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3b}}&{}\approx{}&1.748302881333&{}:{}&1.806562964876&{}:{}&-2.554865846209&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.138384326957&{}:{}&1.334089318960&{}:{}&-0.195704992003&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-2.513669746063&{}:{}&2.513669746063&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.262326850376&{}:{}&1.395639890329&{}:{}&-0.133313039953&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.033897852085&{}:{}&2.102249102308&{}:{}&-1.068351250224&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.004145244698&{}:{}&1.367706141261&{}:{}&-0.363560896563&, \\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.110640395768&{}:{}&1.066624909668&{}:{}&0.044015486100&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.409097417209&{}:{}&0.422730094909&{}:{}&0.168172487882&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.105258927874&{}:{}&1.048560208790&{}:{}&-0.153819136664&, \\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&1.923879532511&{}:{}&-0.923879532511&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.382683432365&{}:{}&0.000000000000&{}:{}&1.382683432365&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.117316567635&{}:{}&0.882683432365&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3b}}}{B^\prime_{\mathbf{3b}}}{C^\prime_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^*_{\mathbf{3b}}}{B^*_{\mathbf{3b}}}{C^*_{\mathbf{3b}}}\)
3b (123)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3b}}}}&{}\approx{}&-2.554865846209&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3b}}}}&{}\approx{}&-0.195704992003&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3b}}}}&{}\approx{}&-3.554865846209&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.707106781187&{}:{}&-0.707106781187&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&1.748302881333&{}:{}&1.806562964876&{}:{}&-2.554865846209&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.138384326957&{}:{}&1.334089318960&{}:{}&-0.195704992003&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-2.513669746063&{}:{}&2.513669746063&{}:{}&1.000000000000&. \end{alignedat} \]
3b (123)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3b}}&{}\approx{}&-0.003362146569&{}:{}&1.109326190582&{}:{}&-0.105964044013&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.262326850376&{}:{}&1.395639890329&{}:{}&-0.133313039953&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.033897852085&{}:{}&2.102249102308&{}:{}&-1.068351250224&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.004145244698&{}:{}&1.367706141261&{}:{}&-0.363560896563&. \end{alignedat} \]
3b (123)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3b}}&{}\approx{}&0.087929062579&{}:{}&0.875924903275&{}:{}&0.036146034145&,\\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.110640395768&{}:{}&1.066624909668&{}:{}&0.044015486100&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.409097417209&{}:{}&0.422730094909&{}:{}&0.168172487882&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.105258927874&{}:{}&1.048560208790&{}:{}&-0.153819136664&. \end{alignedat} \]
3b (123)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3b}}&{}\approx{}&-0.126228645930&{}:{}&1.423455003042&{}:{}&-0.297226357112&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&1.748302881333&{}:{}&1.806562964876&{}:{}&-2.554865846209&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.138384326957&{}:{}&1.334089318960&{}:{}&-0.195704992003&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-2.513669746063&{}:{}&2.513669746063&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.262326850376&{}:{}&1.395639890329&{}:{}&-0.133313039953&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.033897852085&{}:{}&2.102249102308&{}:{}&-1.068351250224&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.004145244698&{}:{}&1.367706141261&{}:{}&-0.363560896563&, \end{alignedat} \]
3b (123)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3b}}&{}\approx{}&0.203632007983&{}:{}&1.532116080188&{}:{}&-0.735748088171&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&1.923879532511&{}:{}&-0.923879532511&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.382683432365&{}:{}&0.000000000000&{}:{}&1.382683432365&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.117316567635&{}:{}&0.882683432365&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3b}}&{}\approx{}&-0.095345022530&{}:{}&1.731886145206&{}:{}&-0.636541122676&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.262326850376&{}:{}&1.395639890329&{}:{}&-0.133313039953&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.033897852085&{}:{}&2.102249102308&{}:{}&-1.068351250224&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.004145244698&{}:{}&1.367706141261&{}:{}&-0.363560896563&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&1.923879532511&{}:{}&-0.923879532511&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.382683432365&{}:{}&0.000000000000&{}:{}&1.382683432365&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.117316567635&{}:{}&0.882683432365&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)