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 > 5a (213)

5a(213)

Malfatti circles

5a (213)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.707106781187&{}:{}&0.707106781187&{}:{}&1.000000000000&, \\ P_{\mathbf{5a}}&{}\approx{}&1.109326190582&{}:{}&-0.003362146569&{}:{}&-0.105964044013&, \\ P^-_{\mathbf{5a}}&{}\approx{}&0.875924903275&{}:{}&0.087929062579&{}:{}&0.036146034145&, \\ P^+_{\mathbf{5a}}&{}\approx{}&1.423455003042&{}:{}&-0.126228645930&{}:{}&-0.297226357112&, \\ Q_{\mathbf{5a}}&{}\approx{}&1.532116080188&{}:{}&0.203632007983&{}:{}&-0.735748088171&, \\ I^\prime_{\mathbf{5a}}&{}\approx{}&1.731886145206&{}:{}&-0.095345022530&{}:{}&-0.636541122676&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{5a}}\)
\(P^-_{\mathbf{5a}}\)
\(P^+_{\mathbf{5a}}\)
\(Q_{\mathbf{5a}}\)
\(I^\prime_{\mathbf{5a}}\)
5a (213)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5a}}&{}\approx{}&1.334089318960&{}:{}&-0.138384326957&{}:{}&-0.195704992003&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&1.806562964876&{}:{}&1.748302881333&{}:{}&-2.554865846209&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&2.513669746063&{}:{}&-2.513669746063&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&2.102249102308&{}:{}&-0.033897852085&{}:{}&-1.068351250224&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.395639890329&{}:{}&-0.262326850376&{}:{}&-0.133313039953&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.367706141261&{}:{}&-0.004145244698&{}:{}&-0.363560896563&, \\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.422730094909&{}:{}&0.409097417209&{}:{}&0.168172487882&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.066624909668&{}:{}&-0.110640395768&{}:{}&0.044015486100&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.048560208790&{}:{}&0.105258927874&{}:{}&-0.153819136664&, \\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.382683432365&{}:{}&1.382683432365&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.923879532511&{}:{}&0.000000000000&{}:{}&-0.923879532511&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.882683432365&{}:{}&0.117316567635&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5a}}}{B^\prime_{\mathbf{5a}}}{C^\prime_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^*_{\mathbf{5a}}}{B^*_{\mathbf{5a}}}{C^*_{\mathbf{5a}}}\)
5a (213)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5a}}}}&{}\approx{}&-0.195704992003&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5a}}}}&{}\approx{}&-2.554865846209&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5a}}}}&{}\approx{}&-3.554865846209&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.707106781187&{}:{}&0.707106781187&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.334089318960&{}:{}&-0.138384326957&{}:{}&-0.195704992003&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&1.806562964876&{}:{}&1.748302881333&{}:{}&-2.554865846209&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&2.513669746063&{}:{}&-2.513669746063&{}:{}&1.000000000000&. \end{alignedat} \]
5a (213)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5a}}&{}\approx{}&1.109326190582&{}:{}&-0.003362146569&{}:{}&-0.105964044013&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&2.102249102308&{}:{}&-0.033897852085&{}:{}&-1.068351250224&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.395639890329&{}:{}&-0.262326850376&{}:{}&-0.133313039953&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.367706141261&{}:{}&-0.004145244698&{}:{}&-0.363560896563&. \end{alignedat} \]
5a (213)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5a}}&{}\approx{}&0.875924903275&{}:{}&0.087929062579&{}:{}&0.036146034145&,\\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.422730094909&{}:{}&0.409097417209&{}:{}&0.168172487882&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.066624909668&{}:{}&-0.110640395768&{}:{}&0.044015486100&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.048560208790&{}:{}&0.105258927874&{}:{}&-0.153819136664&. \end{alignedat} \]
5a (213)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5a}}&{}\approx{}&1.423455003042&{}:{}&-0.126228645930&{}:{}&-0.297226357112&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.334089318960&{}:{}&-0.138384326957&{}:{}&-0.195704992003&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&1.806562964876&{}:{}&1.748302881333&{}:{}&-2.554865846209&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&2.513669746063&{}:{}&-2.513669746063&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&2.102249102308&{}:{}&-0.033897852085&{}:{}&-1.068351250224&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.395639890329&{}:{}&-0.262326850376&{}:{}&-0.133313039953&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.367706141261&{}:{}&-0.004145244698&{}:{}&-0.363560896563&, \end{alignedat} \]
5a (213)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5a}}&{}\approx{}&1.532116080188&{}:{}&0.203632007983&{}:{}&-0.735748088171&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.382683432365&{}:{}&1.382683432365&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.923879532511&{}:{}&0.000000000000&{}:{}&-0.923879532511&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.882683432365&{}:{}&0.117316567635&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5a}}&{}\approx{}&1.731886145206&{}:{}&-0.095345022530&{}:{}&-0.636541122676&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&2.102249102308&{}:{}&-0.033897852085&{}:{}&-1.068351250224&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.395639890329&{}:{}&-0.262326850376&{}:{}&-0.133313039953&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.367706141261&{}:{}&-0.004145244698&{}:{}&-0.363560896563&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.382683432365&{}:{}&1.382683432365&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.923879532511&{}:{}&0.000000000000&{}:{}&-0.923879532511&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.882683432365&{}:{}&0.117316567635&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)