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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1c(112)

Malfatti circles

1c (112)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.100000000000&{}:{}&3.125000000000&{}:{}&-4.225000000000&, \\ P_{\mathbf{1c}}&{}\approx{}&-0.255574043261&{}:{}&-0.570715474210&{}:{}&1.826289517471&, \\ P^-_{\mathbf{1c}}&{}\approx{}&0.114446002805&{}:{}&0.009817671809&{}:{}&0.875736325386&, \\ P^+_{\mathbf{1c}}&{}\approx{}&-0.795092024540&{}:{}&-1.417177914110&{}:{}&3.212269938650&, \\ Q_{\mathbf{1c}}&{}\approx{}&1.636363636364&{}:{}&1.909090909091&{}:{}&-2.545454545455&, \\ I^\prime_{\mathbf{1c}}&{}\approx{}&-1.032258064516&{}:{}&-1.881720430108&{}:{}&3.913978494624&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{1c}}\)
\(P^-_{\mathbf{1c}}\)
\(P^+_{\mathbf{1c}}\)
\(Q_{\mathbf{1c}}\)
\(I^\prime_{\mathbf{1c}}\)
1c (112)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1c}}&{}\approx{}&0.398437500000&{}:{}&-1.708984375000&{}:{}&2.310546875000&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-0.960000000000&{}:{}&0.028571428571&{}:{}&1.931428571429&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-6.000000000000&{}:{}&-8.928571428571&{}:{}&15.928571428571&, \\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.655172413793&{}:{}&-1.206896551724&{}:{}&3.862068965517&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.629508196721&{}:{}&-2.868852459016&{}:{}&4.498360655738&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.341485104491&{}:{}&-0.762561138284&{}:{}&2.104046242775&, \\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.593639575972&{}:{}&0.017667844523&{}:{}&1.575971731449&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.229213483146&{}:{}&-0.983146067416&{}:{}&1.753932584270&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.145247419010&{}:{}&0.012459950160&{}:{}&0.842292630829&, \\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-3.000000000000&{}:{}&4.000000000000&,\\B^*_{\mathbf{1c}}&{}\approx{}&-1.800000000000&{}:{}&0.000000000000&{}:{}&2.800000000000&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.461538461538&{}:{}&0.538461538462&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1c}}}{B^\prime_{\mathbf{1c}}}{C^\prime_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^*_{\mathbf{1c}}}{B^*_{\mathbf{1c}}}{C^*_{\mathbf{1c}}}\)
1c (112)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.546875000000&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.457142857143&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1c}}}}&{}\approx{}&-2.857142857143&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.100000000000&{}:{}&3.125000000000&{}:{}&-4.225000000000&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&0.398437500000&{}:{}&-1.708984375000&{}:{}&2.310546875000&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-0.960000000000&{}:{}&0.028571428571&{}:{}&1.931428571429&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-6.000000000000&{}:{}&-8.928571428571&{}:{}&15.928571428571&. \end{alignedat} \]
1c (112)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1c}}&{}\approx{}&-0.255574043261&{}:{}&-0.570715474210&{}:{}&1.826289517471&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.655172413793&{}:{}&-1.206896551724&{}:{}&3.862068965517&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.629508196721&{}:{}&-2.868852459016&{}:{}&4.498360655738&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.341485104491&{}:{}&-0.762561138284&{}:{}&2.104046242775&. \end{alignedat} \]
1c (112)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1c}}&{}\approx{}&0.114446002805&{}:{}&0.009817671809&{}:{}&0.875736325386&,\\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.593639575972&{}:{}&0.017667844523&{}:{}&1.575971731449&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.229213483146&{}:{}&-0.983146067416&{}:{}&1.753932584270&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.145247419010&{}:{}&0.012459950160&{}:{}&0.842292630829&. \end{alignedat} \]
1c (112)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1c}}&{}\approx{}&-0.795092024540&{}:{}&-1.417177914110&{}:{}&3.212269938650&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&0.398437500000&{}:{}&-1.708984375000&{}:{}&2.310546875000&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-0.960000000000&{}:{}&0.028571428571&{}:{}&1.931428571429&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-6.000000000000&{}:{}&-8.928571428571&{}:{}&15.928571428571&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.655172413793&{}:{}&-1.206896551724&{}:{}&3.862068965517&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.629508196721&{}:{}&-2.868852459016&{}:{}&4.498360655738&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.341485104491&{}:{}&-0.762561138284&{}:{}&2.104046242775&, \end{alignedat} \]
1c (112)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1c}}&{}\approx{}&1.636363636364&{}:{}&1.909090909091&{}:{}&-2.545454545455&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-3.000000000000&{}:{}&4.000000000000&,\\B^*_{\mathbf{1c}}&{}\approx{}&-1.800000000000&{}:{}&0.000000000000&{}:{}&2.800000000000&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.461538461538&{}:{}&0.538461538462&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1c}}&{}\approx{}&-1.032258064516&{}:{}&-1.881720430108&{}:{}&3.913978494624&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.655172413793&{}:{}&-1.206896551724&{}:{}&3.862068965517&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.629508196721&{}:{}&-2.868852459016&{}:{}&4.498360655738&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.341485104491&{}:{}&-0.762561138284&{}:{}&2.104046242775&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-3.000000000000&{}:{}&4.000000000000&,\\B^*_{\mathbf{1c}}&{}\approx{}&-1.800000000000&{}:{}&0.000000000000&{}:{}&2.800000000000&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.461538461538&{}:{}&0.538461538462&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)