Derousseau's Generalization of the Malfatti circles

Ajima's example

不朽算法 十四 (Fukyū Sanpō §14)

\(a=252\),  \(b=375\),  \(c=507\).  \(a:b:c=84:125:169\).

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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&, \\ P_{\mathbf{1}}&{}\approx{}&0.019484712851&{}:{}&0.030403574795&{}:{}&0.950111712354&, \\ P^-_{\mathbf{1}}&{}\approx{}&-0.006145433567&{}:{}&-0.007558465230&{}:{}&1.013703898797&, \\ P^+_{\mathbf{1}}&{}\approx{}&0.039942338879&{}:{}&0.060704346607&{}:{}&0.899353314513&, \\ Q_{\mathbf{1}}&{}\approx{}&-0.074626865672&{}:{}&-0.119402985075&{}:{}&1.194029850746&, \\ I^\prime_{\mathbf{1}}&{}\approx{}&0.080459770115&{}:{}&0.122605363985&{}:{}&0.796934865900&, \end{alignedat} \]
\(I\) Incenter
\(P_{\mathbf{1}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{1}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{1}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{1}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{1}}\) Radical center of the Malfatti circles
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1}}&{}\approx{}&-0.066666666667&{}:{}&0.453514739229&{}:{}&0.613151927438&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.364583333333&{}:{}&-0.098090277778&{}:{}&0.733506944444&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.025925925926&{}:{}&0.038580246914&{}:{}&0.935493827160&, \\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.048409405256&{}:{}&0.029506685108&{}:{}&0.922083909636&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.018666666667&{}:{}&0.071111111111&{}:{}&0.910222222222&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.129687268416&{}:{}&0.202361543402&{}:{}&0.667951188182&, \\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.027163368258&{}:{}&-0.007308239555&{}:{}&0.980144871297&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.005856340385&{}:{}&0.039839050237&{}:{}&0.966017290148&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.143526654950&{}:{}&-0.176528021871&{}:{}&1.320054676821&, \\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.111111111111&{}:{}&1.111111111111&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.066666666667&{}:{}&0.000000000000&{}:{}&1.066666666667&,\\C^*_{\mathbf{1}}&{}\approx{}&0.384615384615&{}:{}&0.615384615385&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1}}}{B^\prime_{\mathbf{1}}}{C^\prime_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1}}}{B^{\prime\prime}_{\mathbf{1}}}{C^{\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1}}}{B^{\prime\prime\prime}_{\mathbf{1}}}{C^{\prime\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^*_{\mathbf{1}}}{B^*_{\mathbf{1}}}{C^*_{\mathbf{1}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1}}}}&{}\approx{}&1.371428571429&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1}}}}&{}\approx{}&1.640625000000&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1}}}}&{}\approx{}&0.116666666667&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.066666666667&{}:{}&0.453514739229&{}:{}&0.613151927438&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.364583333333&{}:{}&-0.098090277778&{}:{}&0.733506944444&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.025925925926&{}:{}&0.038580246914&{}:{}&0.935493827160&. \end{alignedat} \]
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First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1}}&{}\approx{}&0.019484712851&{}:{}&0.030403574795&{}:{}&0.950111712354&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.048409405256&{}:{}&0.029506685108&{}:{}&0.922083909636&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.018666666667&{}:{}&0.071111111111&{}:{}&0.910222222222&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.129687268416&{}:{}&0.202361543402&{}:{}&0.667951188182&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1}}&{}\approx{}&-0.006145433567&{}:{}&-0.007558465230&{}:{}&1.013703898797&,\\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.027163368258&{}:{}&-0.007308239555&{}:{}&0.980144871297&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.005856340385&{}:{}&0.039839050237&{}:{}&0.966017290148&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.143526654950&{}:{}&-0.176528021871&{}:{}&1.320054676821&. \end{alignedat} \]
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Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1}}&{}\approx{}&0.039942338879&{}:{}&0.060704346607&{}:{}&0.899353314513&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.066666666667&{}:{}&0.453514739229&{}:{}&0.613151927438&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.364583333333&{}:{}&-0.098090277778&{}:{}&0.733506944444&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.025925925926&{}:{}&0.038580246914&{}:{}&0.935493827160&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.048409405256&{}:{}&0.029506685108&{}:{}&0.922083909636&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.018666666667&{}:{}&0.071111111111&{}:{}&0.910222222222&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.129687268416&{}:{}&0.202361543402&{}:{}&0.667951188182&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1}}&{}\approx{}&-0.074626865672&{}:{}&-0.119402985075&{}:{}&1.194029850746&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.111111111111&{}:{}&1.111111111111&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.066666666667&{}:{}&0.000000000000&{}:{}&1.066666666667&,\\C^*_{\mathbf{1}}&{}\approx{}&0.384615384615&{}:{}&0.615384615385&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1}}&{}\approx{}&0.080459770115&{}:{}&0.122605363985&{}:{}&0.796934865900&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.048409405256&{}:{}&0.029506685108&{}:{}&0.922083909636&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.018666666667&{}:{}&0.071111111111&{}:{}&0.910222222222&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.129687268416&{}:{}&0.202361543402&{}:{}&0.667951188182&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.111111111111&{}:{}&1.111111111111&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.066666666667&{}:{}&0.000000000000&{}:{}&1.066666666667&,\\C^*_{\mathbf{1}}&{}\approx{}&0.384615384615&{}:{}&0.615384615385&{}:{}&0.000000000000&. \end{alignedat} \]
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