Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 120° Top Angle

\(A=30\degree\), \(B=30\degree\), \(C=120\degree\).


[Top] > Isosceles Triangle with 120° Top Angle > 0b (101)

0b(101)

Malfatti circles

0b (101)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.577350269190&{}:{}&-0.577350269190&{}:{}&1.000000000000&, \\ P_{\mathbf{b}}&{}\approx{}&0.725573314346&{}:{}&-0.297490879250&{}:{}&0.571917564904&, \\ P^-_{\mathbf{b}}&{}\approx{}&0.766318865326&{}:{}&-0.220559353333&{}:{}&0.454240488007&, \\ P^+_{\mathbf{b}}&{}\approx{}&0.699282249043&{}:{}&-0.347130943579&{}:{}&0.647848694536&, \\ Q_{\mathbf{b}}&{}\approx{}&0.849166530897&{}:{}&-0.145693943275&{}:{}&0.296527412378&, \\ I^\prime_{\mathbf{b}}&{}\approx{}&0.654398348313&{}:{}&-0.419023491590&{}:{}&0.764625143278&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{b}}\)
\(P^-_{\mathbf{b}}\)
\(P^+_{\mathbf{b}}\)
\(Q_{\mathbf{b}}\)
\(I^\prime_{\mathbf{b}}\)
0b (101)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{b}}&{}\approx{}&0.868347502413&{}:{}&-0.179840656176&{}:{}&0.311493153763&,\\B^\prime_{\mathbf{b}}&{}\approx{}&0.744962785747&{}:{}&-1.035276180410&{}:{}&1.290313394663&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.349396052863&{}:{}&-0.349396052863&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.475688682152&{}:{}&-0.568377067908&{}:{}&1.092688385756&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.692001894452&{}:{}&-0.237457509424&{}:{}&0.545455614972&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.844353659612&{}:{}&-0.346191773636&{}:{}&0.501838114024&, \\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.404468620352&{}:{}&-0.562090800218&{}:{}&1.157622179866&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.721683111454&{}:{}&-0.149465466250&{}:{}&0.427782354796&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.933791576263&{}:{}&-0.268760793356&{}:{}&0.334969217093&, \\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-0.965925826289&{}:{}&1.965925826289&,\\B^*_{\mathbf{b}}&{}\approx{}&0.741180954897&{}:{}&0.000000000000&{}:{}&0.258819045103&,\\C^*_{\mathbf{b}}&{}\approx{}&1.207106781187&{}:{}&-0.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{b}}}{B^\prime_{\mathbf{b}}}{C^\prime_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{b}}}{B^{\prime\prime}_{\mathbf{b}}}{C^{\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{b}}}{B^{\prime\prime\prime}_{\mathbf{b}}}{C^{\prime\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^*_{\mathbf{b}}}{B^*_{\mathbf{b}}}{C^*_{\mathbf{b}}}\)
0b (101)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{b}}}}&{}\approx{}&0.311493153763&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{b}}}}&{}\approx{}&1.290313394663&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{b}}}}&{}\approx{}&0.605171715523&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.577350269190&{}:{}&-0.577350269190&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.868347502413&{}:{}&-0.179840656176&{}:{}&0.311493153763&,\\B^\prime_{\mathbf{b}}&{}\approx{}&0.744962785747&{}:{}&-1.035276180410&{}:{}&1.290313394663&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.349396052863&{}:{}&-0.349396052863&{}:{}&1.000000000000&. \end{alignedat} \]
0b (101)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{b}}&{}\approx{}&0.725573314346&{}:{}&-0.297490879250&{}:{}&0.571917564904&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.475688682152&{}:{}&-0.568377067908&{}:{}&1.092688385756&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.692001894452&{}:{}&-0.237457509424&{}:{}&0.545455614972&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.844353659612&{}:{}&-0.346191773636&{}:{}&0.501838114024&. \end{alignedat} \]
0b (101)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{b}}&{}\approx{}&0.766318865326&{}:{}&-0.220559353333&{}:{}&0.454240488007&,\\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.404468620352&{}:{}&-0.562090800218&{}:{}&1.157622179866&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.721683111454&{}:{}&-0.149465466250&{}:{}&0.427782354796&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.933791576263&{}:{}&-0.268760793356&{}:{}&0.334969217093&. \end{alignedat} \]
0b (101)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{b}}&{}\approx{}&0.699282249043&{}:{}&-0.347130943579&{}:{}&0.647848694536&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.868347502413&{}:{}&-0.179840656176&{}:{}&0.311493153763&,\\B^\prime_{\mathbf{b}}&{}\approx{}&0.744962785747&{}:{}&-1.035276180410&{}:{}&1.290313394663&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.349396052863&{}:{}&-0.349396052863&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.475688682152&{}:{}&-0.568377067908&{}:{}&1.092688385756&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.692001894452&{}:{}&-0.237457509424&{}:{}&0.545455614972&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.844353659612&{}:{}&-0.346191773636&{}:{}&0.501838114024&, \end{alignedat} \]
0b (101)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{b}}&{}\approx{}&0.849166530897&{}:{}&-0.145693943275&{}:{}&0.296527412378&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-0.965925826289&{}:{}&1.965925826289&,\\B^*_{\mathbf{b}}&{}\approx{}&0.741180954897&{}:{}&0.000000000000&{}:{}&0.258819045103&,\\C^*_{\mathbf{b}}&{}\approx{}&1.207106781187&{}:{}&-0.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{b}}&{}\approx{}&0.654398348313&{}:{}&-0.419023491590&{}:{}&0.764625143278&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.475688682152&{}:{}&-0.568377067908&{}:{}&1.092688385756&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.692001894452&{}:{}&-0.237457509424&{}:{}&0.545455614972&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.844353659612&{}:{}&-0.346191773636&{}:{}&0.501838114024&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-0.965925826289&{}:{}&1.965925826289&,\\B^*_{\mathbf{b}}&{}\approx{}&0.741180954897&{}:{}&0.000000000000&{}:{}&0.258819045103&,\\C^*_{\mathbf{b}}&{}\approx{}&1.207106781187&{}:{}&-0.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)