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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2c(130)

Malfatti circles

2c (130)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.100000000000&{}:{}&3.125000000000&{}:{}&-4.225000000000&, \\ P_{\mathbf{2c}}&{}\approx{}&1.005544330397&{}:{}&0.443546431747&{}:{}&-0.449090762143&, \\ P^-_{\mathbf{2c}}&{}\approx{}&0.908473383194&{}:{}&0.205719313682&{}:{}&-0.114192696876&, \\ P^+_{\mathbf{2c}}&{}\approx{}&1.087990434198&{}:{}&0.645542186683&{}:{}&-0.733532620880&, \\ Q_{\mathbf{2c}}&{}\approx{}&1.148936170213&{}:{}&-0.595744680851&{}:{}&0.446808510638&, \\ I^\prime_{\mathbf{2c}}&{}\approx{}&1.159731543624&{}:{}&0.939597315436&{}:{}&-1.099328859060&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{2c}}\)
\(P^-_{\mathbf{2c}}\)
\(P^+_{\mathbf{2c}}\)
\(Q_{\mathbf{2c}}\)
\(I^\prime_{\mathbf{2c}}\)
2c (130)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2c}}&{}\approx{}&1.120312500000&{}:{}&0.341796875000&{}:{}&-0.462109375000&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&10.800000000000&{}:{}&11.928571428571&{}:{}&-21.728571428571&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.533333333333&{}:{}&0.793650793651&{}:{}&-0.326984126984&, \\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.016470588235&{}:{}&1.317647058824&{}:{}&-1.334117647059&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.943617747440&{}:{}&0.477815699659&{}:{}&-0.421433447099&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.321886952826&{}:{}&0.583085422864&{}:{}&-0.904972375691&, \\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.670509977827&{}:{}&0.740576496674&{}:{}&-0.411086474501&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.847893569845&{}:{}&0.258684405026&{}:{}&-0.106577974871&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.228557829605&{}:{}&0.278200856735&{}:{}&-0.506758686340&, \\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&4.000000000000&{}:{}&-3.000000000000&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.720000000000&{}:{}&0.000000000000&{}:{}&0.280000000000&,\\C^*_{\mathbf{2c}}&{}\approx{}&2.076923076923&{}:{}&-1.076923076923&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2c}}}{B^\prime_{\mathbf{2c}}}{C^\prime_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^*_{\mathbf{2c}}}{B^*_{\mathbf{2c}}}{C^*_{\mathbf{2c}}}\)
2c (130)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2c}}}}&{}\approx{}&0.109375000000&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2c}}}}&{}\approx{}&5.142857142857&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2c}}}}&{}\approx{}&0.253968253968&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.100000000000&{}:{}&3.125000000000&{}:{}&-4.225000000000&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.120312500000&{}:{}&0.341796875000&{}:{}&-0.462109375000&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&10.800000000000&{}:{}&11.928571428571&{}:{}&-21.728571428571&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.533333333333&{}:{}&0.793650793651&{}:{}&-0.326984126984&. \end{alignedat} \]
2c (130)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2c}}&{}\approx{}&1.005544330397&{}:{}&0.443546431747&{}:{}&-0.449090762143&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.016470588235&{}:{}&1.317647058824&{}:{}&-1.334117647059&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.943617747440&{}:{}&0.477815699659&{}:{}&-0.421433447099&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.321886952826&{}:{}&0.583085422864&{}:{}&-0.904972375691&. \end{alignedat} \]
2c (130)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2c}}&{}\approx{}&0.908473383194&{}:{}&0.205719313682&{}:{}&-0.114192696876&,\\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.670509977827&{}:{}&0.740576496674&{}:{}&-0.411086474501&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.847893569845&{}:{}&0.258684405026&{}:{}&-0.106577974871&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.228557829605&{}:{}&0.278200856735&{}:{}&-0.506758686340&. \end{alignedat} \]
2c (130)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2c}}&{}\approx{}&1.087990434198&{}:{}&0.645542186683&{}:{}&-0.733532620880&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.120312500000&{}:{}&0.341796875000&{}:{}&-0.462109375000&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&10.800000000000&{}:{}&11.928571428571&{}:{}&-21.728571428571&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.533333333333&{}:{}&0.793650793651&{}:{}&-0.326984126984&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.016470588235&{}:{}&1.317647058824&{}:{}&-1.334117647059&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.943617747440&{}:{}&0.477815699659&{}:{}&-0.421433447099&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.321886952826&{}:{}&0.583085422864&{}:{}&-0.904972375691&, \end{alignedat} \]
2c (130)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2c}}&{}\approx{}&1.148936170213&{}:{}&-0.595744680851&{}:{}&0.446808510638&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&4.000000000000&{}:{}&-3.000000000000&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.720000000000&{}:{}&0.000000000000&{}:{}&0.280000000000&,\\C^*_{\mathbf{2c}}&{}\approx{}&2.076923076923&{}:{}&-1.076923076923&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2c}}&{}\approx{}&1.159731543624&{}:{}&0.939597315436&{}:{}&-1.099328859060&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.016470588235&{}:{}&1.317647058824&{}:{}&-1.334117647059&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.943617747440&{}:{}&0.477815699659&{}:{}&-0.421433447099&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.321886952826&{}:{}&0.583085422864&{}:{}&-0.904972375691&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&4.000000000000&{}:{}&-3.000000000000&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.720000000000&{}:{}&0.000000000000&{}:{}&0.280000000000&,\\C^*_{\mathbf{2c}}&{}\approx{}&2.076923076923&{}:{}&-1.076923076923&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)