Derousseau's Generalization of the Malfatti circles

Half of an Equilateral Triangle

\(A=30\degree\), \(B=60\degree\), \(C=90\degree\).


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\(\mathbf{3b}\) \((123)\)

Malfatti circles

3b (123)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.788675134595&{}:{}&-1.366025403784&{}:{}&1.577350269190&, \\ P_{\mathbf{3b}}&{}\approx{}&-0.008698014147&{}:{}&1.330033050915&{}:{}&-0.321335036768&, \\ P^-_{\mathbf{3b}}&{}\approx{}&0.111154236119&{}:{}&0.924791573757&{}:{}&-0.035945809875&, \\ P^+_{\mathbf{3b}}&{}\approx{}&-0.180066740870&{}:{}&1.909460769648&{}:{}&-0.729394028778&, \\ Q_{\mathbf{3b}}&{}\approx{}&0.367993549751&{}:{}&1.789811444198&{}:{}&-1.157804993949&, \\ I^\prime_{\mathbf{3b}}&{}\approx{}&-0.149733900418&{}:{}&2.436809690746&{}:{}&-1.287075790329&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{3b}}\)
\(P^-_{\mathbf{3b}}\)
\(P^+_{\mathbf{3b}}\)
\(Q_{\mathbf{3b}}\)
\(I^\prime_{\mathbf{3b}}\)
3b (123)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3b}}&{}\approx{}&1.400099577495&{}:{}&2.586284325104&{}:{}&-2.986383902599&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.208281482962&{}:{}&1.624844448887&{}:{}&-0.416562965925&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-2.331951230074&{}:{}&4.039058011260&{}:{}&-0.707106781187&, \\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.382407688794&{}:{}&1.822793234594&{}:{}&-0.440385545800&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.056752614297&{}:{}&3.153392501810&{}:{}&-2.096639887513&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.011395662887&{}:{}&1.742536632124&{}:{}&-0.731140969236&, \\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.153894226683&{}:{}&1.200558860947&{}:{}&-0.046664634264&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.396223548081&{}:{}&0.731909907060&{}:{}&-0.128133455142&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.139141752801&{}:{}&1.157644773974&{}:{}&-0.296786526774&, \\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&2.831951230074&{}:{}&-1.831951230074&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.465925826289&{}:{}&0.000000000000&{}:{}&1.465925826289&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.170540688701&{}:{}&0.829459311299&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3b}}}{B^\prime_{\mathbf{3b}}}{C^\prime_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^*_{\mathbf{3b}}}{B^*_{\mathbf{3b}}}{C^*_{\mathbf{3b}}}\)
3b (123)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3b}}}}&{}\approx{}&-1.893291528795&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3b}}}}&{}\approx{}&-0.264090338120&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3b}}}}&{}\approx{}&-2.956795678960&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.788675134595&{}:{}&-1.366025403784&{}:{}&1.577350269190&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&1.400099577495&{}:{}&2.586284325104&{}:{}&-2.986383902599&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.208281482962&{}:{}&1.624844448887&{}:{}&-0.416562965925&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-2.331951230074&{}:{}&4.039058011260&{}:{}&-0.707106781187&. \end{alignedat} \]
3b (123)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3b}}&{}\approx{}&-0.008698014147&{}:{}&1.330033050915&{}:{}&-0.321335036768&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.382407688794&{}:{}&1.822793234594&{}:{}&-0.440385545800&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.056752614297&{}:{}&3.153392501810&{}:{}&-2.096639887513&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.011395662887&{}:{}&1.742536632124&{}:{}&-0.731140969236&. \end{alignedat} \]
3b (123)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3b}}&{}\approx{}&0.111154236119&{}:{}&0.924791573757&{}:{}&-0.035945809875&,\\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.153894226683&{}:{}&1.200558860947&{}:{}&-0.046664634264&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.396223548081&{}:{}&0.731909907060&{}:{}&-0.128133455142&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.139141752801&{}:{}&1.157644773974&{}:{}&-0.296786526774&. \end{alignedat} \]
3b (123)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3b}}&{}\approx{}&-0.180066740870&{}:{}&1.909460769648&{}:{}&-0.729394028778&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&1.400099577495&{}:{}&2.586284325104&{}:{}&-2.986383902599&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.208281482962&{}:{}&1.624844448887&{}:{}&-0.416562965925&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-2.331951230074&{}:{}&4.039058011260&{}:{}&-0.707106781187&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.382407688794&{}:{}&1.822793234594&{}:{}&-0.440385545800&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.056752614297&{}:{}&3.153392501810&{}:{}&-2.096639887513&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.011395662887&{}:{}&1.742536632124&{}:{}&-0.731140969236&, \end{alignedat} \]
3b (123)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3b}}&{}\approx{}&0.367993549751&{}:{}&1.789811444198&{}:{}&-1.157804993949&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&2.831951230074&{}:{}&-1.831951230074&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.465925826289&{}:{}&0.000000000000&{}:{}&1.465925826289&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.170540688701&{}:{}&0.829459311299&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3b}}&{}\approx{}&-0.149733900418&{}:{}&2.436809690746&{}:{}&-1.287075790329&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.382407688794&{}:{}&1.822793234594&{}:{}&-0.440385545800&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.056752614297&{}:{}&3.153392501810&{}:{}&-2.096639887513&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.011395662887&{}:{}&1.742536632124&{}:{}&-0.731140969236&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&2.831951230074&{}:{}&-1.831951230074&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.465925826289&{}:{}&0.000000000000&{}:{}&1.465925826289&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.170540688701&{}:{}&0.829459311299&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)

Hiroyasu Kamo