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 > 5b (303)

5b(303)

Malfatti circles

5b (303)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.707106781187&{}:{}&-0.707106781187&{}:{}&1.000000000000&, \\ P_{\mathbf{5b}}&{}\approx{}&0.139290338881&{}:{}&-0.014341055345&{}:{}&0.875050716464&, \\ P^-_{\mathbf{5b}}&{}\approx{}&0.070611680304&{}:{}&0.069450496314&{}:{}&0.859937823381&, \\ P^+_{\mathbf{5b}}&{}\approx{}&0.194591420426&{}:{}&-0.081811264100&{}:{}&0.887219843673&, \\ Q_{\mathbf{5b}}&{}\approx{}&0.364161428209&{}:{}&0.048400329332&{}:{}&0.587438242459&, \\ I^\prime_{\mathbf{5b}}&{}\approx{}&0.273239171748&{}:{}&-0.087674443642&{}:{}&0.814435271894&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{5b}}\)
\(P^-_{\mathbf{5b}}\)
\(P^+_{\mathbf{5b}}\)
\(Q_{\mathbf{5b}}\)
\(I^\prime_{\mathbf{5b}}\)
5b (303)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5b}}&{}\approx{}&0.665910681040&{}:{}&-0.806562964876&{}:{}&1.140652283836&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.309957202211&{}:{}&0.251697118667&{}:{}&0.438345679122&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.099456183690&{}:{}&-0.099456183690&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.150591854105&{}:{}&-0.014152750668&{}:{}&0.863560896563&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.161637432982&{}:{}&-0.177077216346&{}:{}&1.015439783364&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.408773459783&{}:{}&-0.042086499736&{}:{}&0.633313039953&, \\ A^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.084269303489&{}:{}&0.068429901706&{}:{}&0.847300794806&,\\B^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.083561807694&{}:{}&-0.101211560774&{}:{}&1.017649753080&,\\C^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.270527915470&{}:{}&0.266079180035&{}:{}&0.463392904496&, \\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.076120467489&{}:{}&0.923879532511&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.382683432365&{}:{}&0.000000000000&{}:{}&0.617316567635&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.882683432365&{}:{}&0.117316567635&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5b}}}{B^\prime_{\mathbf{5b}}}{C^\prime_{\mathbf{5b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5b}}}{B^{\prime\prime}_{\mathbf{5b}}}{C^{\prime\prime}_{\mathbf{5b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5b}}}{B^{\prime\prime\prime}_{\mathbf{5b}}}{C^{\prime\prime\prime}_{\mathbf{5b}}}\)
\(\triangle{A^*_{\mathbf{5b}}}{B^*_{\mathbf{5b}}}{C^*_{\mathbf{5b}}}\)
5b (303)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5b}}}}&{}\approx{}&1.140652283836&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5b}}}}&{}\approx{}&0.438345679122&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5b}}}}&{}\approx{}&0.140652283836&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.707106781187&{}:{}&-0.707106781187&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{5b}}&{}\approx{}&0.665910681040&{}:{}&-0.806562964876&{}:{}&1.140652283836&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.309957202211&{}:{}&0.251697118667&{}:{}&0.438345679122&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.099456183690&{}:{}&-0.099456183690&{}:{}&1.000000000000&. \end{alignedat} \]
5b (303)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5b}}&{}\approx{}&0.139290338881&{}:{}&-0.014341055345&{}:{}&0.875050716464&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.150591854105&{}:{}&-0.014152750668&{}:{}&0.863560896563&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.161637432982&{}:{}&-0.177077216346&{}:{}&1.015439783364&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.408773459783&{}:{}&-0.042086499736&{}:{}&0.633313039953&. \end{alignedat} \]
5b (303)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5b}}&{}\approx{}&0.070611680304&{}:{}&0.069450496314&{}:{}&0.859937823381&,\\ A^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.084269303489&{}:{}&0.068429901706&{}:{}&0.847300794806&,\\B^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.083561807694&{}:{}&-0.101211560774&{}:{}&1.017649753080&,\\C^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.270527915470&{}:{}&0.266079180035&{}:{}&0.463392904496&. \end{alignedat} \]
5b (303)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5b}}&{}\approx{}&0.194591420426&{}:{}&-0.081811264100&{}:{}&0.887219843673&,\\ A^\prime_{\mathbf{5b}}&{}\approx{}&0.665910681040&{}:{}&-0.806562964876&{}:{}&1.140652283836&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.309957202211&{}:{}&0.251697118667&{}:{}&0.438345679122&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.099456183690&{}:{}&-0.099456183690&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.150591854105&{}:{}&-0.014152750668&{}:{}&0.863560896563&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.161637432982&{}:{}&-0.177077216346&{}:{}&1.015439783364&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.408773459783&{}:{}&-0.042086499736&{}:{}&0.633313039953&, \end{alignedat} \]
5b (303)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5b}}&{}\approx{}&0.364161428209&{}:{}&0.048400329332&{}:{}&0.587438242459&,\\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.076120467489&{}:{}&0.923879532511&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.382683432365&{}:{}&0.000000000000&{}:{}&0.617316567635&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.882683432365&{}:{}&0.117316567635&{}:{}&0.000000000000&. \end{alignedat} \]
5b (303)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5b}}&{}\approx{}&0.273239171748&{}:{}&-0.087674443642&{}:{}&0.814435271894&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.150591854105&{}:{}&-0.014152750668&{}:{}&0.863560896563&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.161637432982&{}:{}&-0.177077216346&{}:{}&1.015439783364&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.408773459783&{}:{}&-0.042086499736&{}:{}&0.633313039953&,\\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.076120467489&{}:{}&0.923879532511&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.382683432365&{}:{}&0.000000000000&{}:{}&0.617316567635&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.882683432365&{}:{}&0.117316567635&{}:{}&0.000000000000&. \end{alignedat} \]
5b (303)