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 > 0a (011)

0a(011)

Malfatti circles

0a (011)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.577350269190&{}:{}&0.577350269190&{}:{}&1.000000000000&, \\ P_{\mathbf{a}}&{}\approx{}&-0.297490879250&{}:{}&0.725573314346&{}:{}&0.571917564904&, \\ P^-_{\mathbf{a}}&{}\approx{}&-0.220559353333&{}:{}&0.766318865326&{}:{}&0.454240488007&, \\ P^+_{\mathbf{a}}&{}\approx{}&-0.347130943579&{}:{}&0.699282249043&{}:{}&0.647848694536&, \\ Q_{\mathbf{a}}&{}\approx{}&-0.145693943275&{}:{}&0.849166530897&{}:{}&0.296527412378&, \\ I^\prime_{\mathbf{a}}&{}\approx{}&-0.419023491590&{}:{}&0.654398348313&{}:{}&0.764625143278&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{a}}\)
\(P^-_{\mathbf{a}}\)
\(P^+_{\mathbf{a}}\)
\(Q_{\mathbf{a}}\)
\(I^\prime_{\mathbf{a}}\)
0a (011)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{a}}&{}\approx{}&-1.035276180410&{}:{}&0.744962785747&{}:{}&1.290313394663&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.179840656176&{}:{}&0.868347502413&{}:{}&0.311493153763&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.349396052863&{}:{}&0.349396052863&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.237457509424&{}:{}&0.692001894452&{}:{}&0.545455614972&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.568377067908&{}:{}&0.475688682152&{}:{}&1.092688385756&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.346191773636&{}:{}&0.844353659612&{}:{}&0.501838114024&, \\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.149465466250&{}:{}&0.721683111454&{}:{}&0.427782354796&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.562090800218&{}:{}&0.404468620352&{}:{}&1.157622179866&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.268760793356&{}:{}&0.933791576263&{}:{}&0.334969217093&, \\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.741180954897&{}:{}&0.258819045103&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.965925826289&{}:{}&0.000000000000&{}:{}&1.965925826289&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.207106781187&{}:{}&1.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{a}}}{B^\prime_{\mathbf{a}}}{C^\prime_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{a}}}{B^{\prime\prime}_{\mathbf{a}}}{C^{\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{a}}}{B^{\prime\prime\prime}_{\mathbf{a}}}{C^{\prime\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^*_{\mathbf{a}}}{B^*_{\mathbf{a}}}{C^*_{\mathbf{a}}}\)
0a (011)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{a}}}}&{}\approx{}&1.290313394663&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{a}}}}&{}\approx{}&0.311493153763&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{a}}}}&{}\approx{}&0.605171715523&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.577350269190&{}:{}&0.577350269190&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-1.035276180410&{}:{}&0.744962785747&{}:{}&1.290313394663&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.179840656176&{}:{}&0.868347502413&{}:{}&0.311493153763&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.349396052863&{}:{}&0.349396052863&{}:{}&1.000000000000&. \end{alignedat} \]
0a (011)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{a}}&{}\approx{}&-0.297490879250&{}:{}&0.725573314346&{}:{}&0.571917564904&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.237457509424&{}:{}&0.692001894452&{}:{}&0.545455614972&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.568377067908&{}:{}&0.475688682152&{}:{}&1.092688385756&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.346191773636&{}:{}&0.844353659612&{}:{}&0.501838114024&. \end{alignedat} \]
0a (011)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{a}}&{}\approx{}&-0.220559353333&{}:{}&0.766318865326&{}:{}&0.454240488007&,\\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.149465466250&{}:{}&0.721683111454&{}:{}&0.427782354796&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.562090800218&{}:{}&0.404468620352&{}:{}&1.157622179866&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.268760793356&{}:{}&0.933791576263&{}:{}&0.334969217093&. \end{alignedat} \]
0a (011)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{a}}&{}\approx{}&-0.347130943579&{}:{}&0.699282249043&{}:{}&0.647848694536&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-1.035276180410&{}:{}&0.744962785747&{}:{}&1.290313394663&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.179840656176&{}:{}&0.868347502413&{}:{}&0.311493153763&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.349396052863&{}:{}&0.349396052863&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.237457509424&{}:{}&0.692001894452&{}:{}&0.545455614972&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.568377067908&{}:{}&0.475688682152&{}:{}&1.092688385756&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.346191773636&{}:{}&0.844353659612&{}:{}&0.501838114024&, \end{alignedat} \]
0a (011)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{a}}&{}\approx{}&-0.145693943275&{}:{}&0.849166530897&{}:{}&0.296527412378&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.741180954897&{}:{}&0.258819045103&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.965925826289&{}:{}&0.000000000000&{}:{}&1.965925826289&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.207106781187&{}:{}&1.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{a}}&{}\approx{}&-0.419023491590&{}:{}&0.654398348313&{}:{}&0.764625143278&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.237457509424&{}:{}&0.692001894452&{}:{}&0.545455614972&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.568377067908&{}:{}&0.475688682152&{}:{}&1.092688385756&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.346191773636&{}:{}&0.844353659612&{}:{}&0.501838114024&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.741180954897&{}:{}&0.258819045103&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.965925826289&{}:{}&0.000000000000&{}:{}&1.965925826289&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.207106781187&{}:{}&1.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)