Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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5a(213)

Malfatti circles

5a (213)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.250000000000&{}:{}&0.600000000000&{}:{}&0.650000000000&, \\ P_{\mathbf{5a}}&{}\approx{}&1.011944825222&{}:{}&-0.000378916047&{}:{}&-0.011565909174&, \\ P^-_{\mathbf{5a}}&{}\approx{}&0.911679386795&{}:{}&0.047323054781&{}:{}&0.040997558424&, \\ P^+_{\mathbf{5a}}&{}\approx{}&1.131153218295&{}:{}&-0.057093127625&{}:{}&-0.074060090670&, \\ Q_{\mathbf{5a}}&{}\approx{}&1.228592719729&{}:{}&0.065107594704&{}:{}&-0.293700314433&, \\ I^\prime_{\mathbf{5a}}&{}\approx{}&1.253701868665&{}:{}&-0.037583132594&{}:{}&-0.216118736071&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{5a}}\)
\(P^-_{\mathbf{5a}}\)
\(P^+_{\mathbf{5a}}\)
\(Q_{\mathbf{5a}}\)
\(I^\prime_{\mathbf{5a}}\)
5a (213)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5a}}&{}\approx{}&1.122256164685&{}:{}&-0.058682959049&{}:{}&-0.063573205636&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&1.278054161137&{}:{}&3.044886657820&{}:{}&-3.322940818957&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.506052104307&{}:{}&-3.614525050336&{}:{}&3.108472946029&, \\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.382717450616&{}:{}&-0.012140636710&{}:{}&-0.370576813905&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.128387666454&{}:{}&-0.115490886967&{}:{}&-0.012896779487&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.125180876643&{}:{}&-0.000421316538&{}:{}&-0.124759560106&, \\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.183606606791&{}:{}&0.437431624031&{}:{}&0.378961769178&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.007374882739&{}:{}&-0.052675797960&{}:{}&0.045300915221&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.004762690979&{}:{}&0.052154782214&{}:{}&-0.056917473193&, \\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.284819196258&{}:{}&1.284819196258&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.314154134496&{}:{}&0.000000000000&{}:{}&-0.314154134496&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.949673356358&{}:{}&0.050326643642&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5a}}}{B^\prime_{\mathbf{5a}}}{C^\prime_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^*_{\mathbf{5a}}}{B^*_{\mathbf{5a}}}{C^*_{\mathbf{5a}}}\)
5a (213)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5a}}}}&{}\approx{}&-0.097804931748&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5a}}}}&{}\approx{}&-5.112216644549&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5a}}}}&{}\approx{}&-6.024208417227&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.250000000000&{}:{}&0.600000000000&{}:{}&0.650000000000&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.122256164685&{}:{}&-0.058682959049&{}:{}&-0.063573205636&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&1.278054161137&{}:{}&3.044886657820&{}:{}&-3.322940818957&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.506052104307&{}:{}&-3.614525050336&{}:{}&3.108472946029&. \end{alignedat} \]
5a (213)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5a}}&{}\approx{}&1.011944825222&{}:{}&-0.000378916047&{}:{}&-0.011565909174&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.382717450616&{}:{}&-0.012140636710&{}:{}&-0.370576813905&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.128387666454&{}:{}&-0.115490886967&{}:{}&-0.012896779487&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.125180876643&{}:{}&-0.000421316538&{}:{}&-0.124759560106&. \end{alignedat} \]
5a (213)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5a}}&{}\approx{}&0.911679386795&{}:{}&0.047323054781&{}:{}&0.040997558424&,\\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.183606606791&{}:{}&0.437431624031&{}:{}&0.378961769178&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.007374882739&{}:{}&-0.052675797960&{}:{}&0.045300915221&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.004762690979&{}:{}&0.052154782214&{}:{}&-0.056917473193&. \end{alignedat} \]
5a (213)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5a}}&{}\approx{}&1.131153218295&{}:{}&-0.057093127625&{}:{}&-0.074060090670&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.122256164685&{}:{}&-0.058682959049&{}:{}&-0.063573205636&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&1.278054161137&{}:{}&3.044886657820&{}:{}&-3.322940818957&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.506052104307&{}:{}&-3.614525050336&{}:{}&3.108472946029&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.382717450616&{}:{}&-0.012140636710&{}:{}&-0.370576813905&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.128387666454&{}:{}&-0.115490886967&{}:{}&-0.012896779487&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.125180876643&{}:{}&-0.000421316538&{}:{}&-0.124759560106&, \end{alignedat} \]
5a (213)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5a}}&{}\approx{}&1.228592719729&{}:{}&0.065107594704&{}:{}&-0.293700314433&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.284819196258&{}:{}&1.284819196258&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.314154134496&{}:{}&0.000000000000&{}:{}&-0.314154134496&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.949673356358&{}:{}&0.050326643642&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5a}}&{}\approx{}&1.253701868665&{}:{}&-0.037583132594&{}:{}&-0.216118736071&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.382717450616&{}:{}&-0.012140636710&{}:{}&-0.370576813905&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.128387666454&{}:{}&-0.115490886967&{}:{}&-0.012896779487&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.125180876643&{}:{}&-0.000421316538&{}:{}&-0.124759560106&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.284819196258&{}:{}&1.284819196258&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.314154134496&{}:{}&0.000000000000&{}:{}&-0.314154134496&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.949673356358&{}:{}&0.050326643642&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)