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 > 3a (033)

3a(033)

Malfatti circles

3a (033)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.707106781187&{}:{}&0.707106781187&{}:{}&1.000000000000&, \\ P_{\mathbf{3a}}&{}\approx{}&-0.014341055345&{}:{}&0.139290338881&{}:{}&0.875050716464&, \\ P^-_{\mathbf{3a}}&{}\approx{}&0.069450496314&{}:{}&0.070611680304&{}:{}&0.859937823381&, \\ P^+_{\mathbf{3a}}&{}\approx{}&-0.081811264100&{}:{}&0.194591420426&{}:{}&0.887219843673&, \\ Q_{\mathbf{3a}}&{}\approx{}&0.048400329332&{}:{}&0.364161428209&{}:{}&0.587438242459&, \\ I^\prime_{\mathbf{3a}}&{}\approx{}&-0.087674443642&{}:{}&0.273239171748&{}:{}&0.814435271894&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{3a}}\)
\(P^-_{\mathbf{3a}}\)
\(P^+_{\mathbf{3a}}\)
\(Q_{\mathbf{3a}}\)
\(I^\prime_{\mathbf{3a}}\)
3a (033)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3a}}&{}\approx{}&0.251697118667&{}:{}&0.309957202211&{}:{}&0.438345679122&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.806562964876&{}:{}&0.665910681040&{}:{}&1.140652283836&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.099456183690&{}:{}&0.099456183690&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.177077216346&{}:{}&0.161637432982&{}:{}&1.015439783364&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.014152750668&{}:{}&0.150591854105&{}:{}&0.863560896563&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.042086499736&{}:{}&0.408773459783&{}:{}&0.633313039953&, \\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.101211560774&{}:{}&0.083561807694&{}:{}&1.017649753080&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.068429901706&{}:{}&0.084269303489&{}:{}&0.847300794806&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.266079180035&{}:{}&0.270527915470&{}:{}&0.463392904496&, \\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.382683432365&{}:{}&0.617316567635&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.076120467489&{}:{}&0.000000000000&{}:{}&0.923879532511&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.117316567635&{}:{}&0.882683432365&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3a}}}{B^\prime_{\mathbf{3a}}}{C^\prime_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^*_{\mathbf{3a}}}{B^*_{\mathbf{3a}}}{C^*_{\mathbf{3a}}}\)
3a (033)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3a}}}}&{}\approx{}&0.438345679122&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3a}}}}&{}\approx{}&1.140652283836&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3a}}}}&{}\approx{}&0.140652283836&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.707106781187&{}:{}&0.707106781187&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.251697118667&{}:{}&0.309957202211&{}:{}&0.438345679122&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.806562964876&{}:{}&0.665910681040&{}:{}&1.140652283836&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.099456183690&{}:{}&0.099456183690&{}:{}&1.000000000000&. \end{alignedat} \]
3a (033)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3a}}&{}\approx{}&-0.014341055345&{}:{}&0.139290338881&{}:{}&0.875050716464&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.177077216346&{}:{}&0.161637432982&{}:{}&1.015439783364&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.014152750668&{}:{}&0.150591854105&{}:{}&0.863560896563&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.042086499736&{}:{}&0.408773459783&{}:{}&0.633313039953&. \end{alignedat} \]
3a (033)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3a}}&{}\approx{}&0.069450496314&{}:{}&0.070611680304&{}:{}&0.859937823381&,\\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.101211560774&{}:{}&0.083561807694&{}:{}&1.017649753080&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.068429901706&{}:{}&0.084269303489&{}:{}&0.847300794806&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.266079180035&{}:{}&0.270527915470&{}:{}&0.463392904496&. \end{alignedat} \]
3a (033)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3a}}&{}\approx{}&-0.081811264100&{}:{}&0.194591420426&{}:{}&0.887219843673&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.251697118667&{}:{}&0.309957202211&{}:{}&0.438345679122&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.806562964876&{}:{}&0.665910681040&{}:{}&1.140652283836&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.099456183690&{}:{}&0.099456183690&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.177077216346&{}:{}&0.161637432982&{}:{}&1.015439783364&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.014152750668&{}:{}&0.150591854105&{}:{}&0.863560896563&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.042086499736&{}:{}&0.408773459783&{}:{}&0.633313039953&, \end{alignedat} \]
3a (033)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3a}}&{}\approx{}&0.048400329332&{}:{}&0.364161428209&{}:{}&0.587438242459&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.382683432365&{}:{}&0.617316567635&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.076120467489&{}:{}&0.000000000000&{}:{}&0.923879532511&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.117316567635&{}:{}&0.882683432365&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3a}}&{}\approx{}&-0.087674443642&{}:{}&0.273239171748&{}:{}&0.814435271894&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.177077216346&{}:{}&0.161637432982&{}:{}&1.015439783364&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.014152750668&{}:{}&0.150591854105&{}:{}&0.863560896563&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.042086499736&{}:{}&0.408773459783&{}:{}&0.633313039953&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.382683432365&{}:{}&0.617316567635&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.076120467489&{}:{}&0.000000000000&{}:{}&0.923879532511&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.117316567635&{}:{}&0.882683432365&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)