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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6a(231)

Malfatti circles

6a (231)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.400000000000&{}:{}&0.595238095238&{}:{}&0.804761904762&, \\ P_{\mathbf{6a}}&{}\approx{}&1.004326767840&{}:{}&-0.003736334702&{}:{}&-0.000590433138&, \\ P^-_{\mathbf{6a}}&{}\approx{}&0.957004680187&{}:{}&0.016447515043&{}:{}&0.026547804769&, \\ P^+_{\mathbf{6a}}&{}\approx{}&1.055068584811&{}:{}&-0.025378769775&{}:{}&-0.029689815036&, \\ Q_{\mathbf{6a}}&{}\approx{}&1.170731707317&{}:{}&-0.219512195122&{}:{}&0.048780487805&, \\ I^\prime_{\mathbf{6a}}&{}\approx{}&1.122077922078&{}:{}&-0.083487940631&{}:{}&-0.038589981447&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{6a}}\)
\(P^-_{\mathbf{6a}}\)
\(P^+_{\mathbf{6a}}\)
\(Q_{\mathbf{6a}}\)
\(I^\prime_{\mathbf{6a}}\)
6a (231)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6a}}&{}\approx{}&1.051851851852&{}:{}&-0.022045855379&{}:{}&-0.029805996473&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&4.800000000000&{}:{}&5.857142857143&{}:{}&-9.657142857143&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.675000000000&{}:{}&-1.004464285714&{}:{}&1.329464285714&, \\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.183561643836&{}:{}&-0.158512720157&{}:{}&-0.025048923679&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.043758389262&{}:{}&-0.043144774688&{}:{}&-0.000613614573&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.063384615385&{}:{}&-0.003956043956&{}:{}&-0.059428571429&, \\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.238674033149&{}:{}&0.291239147593&{}:{}&0.470086819258&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.993264248705&{}:{}&-0.020817912657&{}:{}&0.027553663953&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.011275964392&{}:{}&0.017380245867&{}:{}&-0.028656210259&, \\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.285714285714&{}:{}&-0.285714285714&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.960000000000&{}:{}&0.000000000000&{}:{}&0.040000000000&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.230769230769&{}:{}&-0.230769230769&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6a}}}{B^\prime_{\mathbf{6a}}}{C^\prime_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^*_{\mathbf{6a}}}{B^*_{\mathbf{6a}}}{C^*_{\mathbf{6a}}}\)
6a (231)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6a}}}}&{}\approx{}&-0.037037037037&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6a}}}}&{}\approx{}&-12.000000000000&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6a}}}}&{}\approx{}&-1.687500000000&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.400000000000&{}:{}&0.595238095238&{}:{}&0.804761904762&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.051851851852&{}:{}&-0.022045855379&{}:{}&-0.029805996473&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&4.800000000000&{}:{}&5.857142857143&{}:{}&-9.657142857143&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.675000000000&{}:{}&-1.004464285714&{}:{}&1.329464285714&. \end{alignedat} \]
6a (231)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6a}}&{}\approx{}&1.004326767840&{}:{}&-0.003736334702&{}:{}&-0.000590433138&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.183561643836&{}:{}&-0.158512720157&{}:{}&-0.025048923679&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.043758389262&{}:{}&-0.043144774688&{}:{}&-0.000613614573&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.063384615385&{}:{}&-0.003956043956&{}:{}&-0.059428571429&. \end{alignedat} \]
6a (231)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6a}}&{}\approx{}&0.957004680187&{}:{}&0.016447515043&{}:{}&0.026547804769&,\\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.238674033149&{}:{}&0.291239147593&{}:{}&0.470086819258&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.993264248705&{}:{}&-0.020817912657&{}:{}&0.027553663953&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.011275964392&{}:{}&0.017380245867&{}:{}&-0.028656210259&. \end{alignedat} \]
6a (231)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6a}}&{}\approx{}&1.055068584811&{}:{}&-0.025378769775&{}:{}&-0.029689815036&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.051851851852&{}:{}&-0.022045855379&{}:{}&-0.029805996473&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&4.800000000000&{}:{}&5.857142857143&{}:{}&-9.657142857143&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.675000000000&{}:{}&-1.004464285714&{}:{}&1.329464285714&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.183561643836&{}:{}&-0.158512720157&{}:{}&-0.025048923679&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.043758389262&{}:{}&-0.043144774688&{}:{}&-0.000613614573&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.063384615385&{}:{}&-0.003956043956&{}:{}&-0.059428571429&, \end{alignedat} \]
6a (231)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6a}}&{}\approx{}&1.170731707317&{}:{}&-0.219512195122&{}:{}&0.048780487805&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.285714285714&{}:{}&-0.285714285714&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.960000000000&{}:{}&0.000000000000&{}:{}&0.040000000000&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.230769230769&{}:{}&-0.230769230769&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6a}}&{}\approx{}&1.122077922078&{}:{}&-0.083487940631&{}:{}&-0.038589981447&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.183561643836&{}:{}&-0.158512720157&{}:{}&-0.025048923679&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.043758389262&{}:{}&-0.043144774688&{}:{}&-0.000613614573&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.063384615385&{}:{}&-0.003956043956&{}:{}&-0.059428571429&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.285714285714&{}:{}&-0.285714285714&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.960000000000&{}:{}&0.000000000000&{}:{}&0.040000000000&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.230769230769&{}:{}&-0.230769230769&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)