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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1b(103)

Malfatti circles

1b (103)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.656250000000&{}:{}&-0.976562500000&{}:{}&1.320312500000&, \\ P_{\mathbf{1b}}&{}\approx{}&0.889510366639&{}:{}&-0.203401370505&{}:{}&0.313891003866&, \\ P^-_{\mathbf{1b}}&{}\approx{}&0.929898996839&{}:{}&-0.069529854336&{}:{}&0.139630857496&, \\ P^+_{\mathbf{1b}}&{}\approx{}&0.859510564459&{}:{}&-0.302838241298&{}:{}&0.443327676840&, \\ Q_{\mathbf{1b}}&{}\approx{}&1.115702479339&{}:{}&0.173553719008&{}:{}&-0.289256198347&, \\ I^\prime_{\mathbf{1b}}&{}\approx{}&0.794117647059&{}:{}&-0.463235294118&{}:{}&0.669117647059&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{1b}}\)
\(P^-_{\mathbf{1b}}\)
\(P^+_{\mathbf{1b}}\)
\(Q_{\mathbf{1b}}\)
\(I^\prime_{\mathbf{1b}}\)
1b (103)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1b}}&{}\approx{}&0.939843750000&{}:{}&-0.170898437500&{}:{}&0.231054687500&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.416666666667&{}:{}&-0.254960317460&{}:{}&0.838293650794&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.350000000000&{}:{}&-2.008928571429&{}:{}&1.658928571429&, \\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.636792452830&{}:{}&-0.668632075472&{}:{}&1.031839622642&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.972000000000&{}:{}&-0.315000000000&{}:{}&0.343000000000&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.826800587947&{}:{}&-0.189061734444&{}:{}&0.362261146497&, \\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.618455497382&{}:{}&-0.378435863874&{}:{}&0.759980366492&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.032716520412&{}:{}&-0.187786150324&{}:{}&0.155069629912&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.853918228279&{}:{}&-0.063848665531&{}:{}&0.209930437252&, \\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-1.500000000000&{}:{}&2.500000000000&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.350000000000&{}:{}&0.000000000000&{}:{}&-0.350000000000&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.865384615385&{}:{}&0.134615384615&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1b}}}{B^\prime_{\mathbf{1b}}}{C^\prime_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^*_{\mathbf{1b}}}{B^*_{\mathbf{1b}}}{C^*_{\mathbf{1b}}}\)
1b (103)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1b}}}}&{}\approx{}&0.175000000000&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1b}}}}&{}\approx{}&0.634920634921&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1b}}}}&{}\approx{}&2.057142857143&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.656250000000&{}:{}&-0.976562500000&{}:{}&1.320312500000&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.939843750000&{}:{}&-0.170898437500&{}:{}&0.231054687500&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.416666666667&{}:{}&-0.254960317460&{}:{}&0.838293650794&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.350000000000&{}:{}&-2.008928571429&{}:{}&1.658928571429&. \end{alignedat} \]
1b (103)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1b}}&{}\approx{}&0.889510366639&{}:{}&-0.203401370505&{}:{}&0.313891003866&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.636792452830&{}:{}&-0.668632075472&{}:{}&1.031839622642&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.972000000000&{}:{}&-0.315000000000&{}:{}&0.343000000000&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.826800587947&{}:{}&-0.189061734444&{}:{}&0.362261146497&. \end{alignedat} \]
1b (103)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1b}}&{}\approx{}&0.929898996839&{}:{}&-0.069529854336&{}:{}&0.139630857496&,\\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.618455497382&{}:{}&-0.378435863874&{}:{}&0.759980366492&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.032716520412&{}:{}&-0.187786150324&{}:{}&0.155069629912&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.853918228279&{}:{}&-0.063848665531&{}:{}&0.209930437252&. \end{alignedat} \]
1b (103)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1b}}&{}\approx{}&0.859510564459&{}:{}&-0.302838241298&{}:{}&0.443327676840&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.939843750000&{}:{}&-0.170898437500&{}:{}&0.231054687500&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.416666666667&{}:{}&-0.254960317460&{}:{}&0.838293650794&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.350000000000&{}:{}&-2.008928571429&{}:{}&1.658928571429&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.636792452830&{}:{}&-0.668632075472&{}:{}&1.031839622642&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.972000000000&{}:{}&-0.315000000000&{}:{}&0.343000000000&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.826800587947&{}:{}&-0.189061734444&{}:{}&0.362261146497&, \end{alignedat} \]
1b (103)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1b}}&{}\approx{}&1.115702479339&{}:{}&0.173553719008&{}:{}&-0.289256198347&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-1.500000000000&{}:{}&2.500000000000&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.350000000000&{}:{}&0.000000000000&{}:{}&-0.350000000000&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.865384615385&{}:{}&0.134615384615&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1b}}&{}\approx{}&0.794117647059&{}:{}&-0.463235294118&{}:{}&0.669117647059&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.636792452830&{}:{}&-0.668632075472&{}:{}&1.031839622642&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.972000000000&{}:{}&-0.315000000000&{}:{}&0.343000000000&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.826800587947&{}:{}&-0.189061734444&{}:{}&0.362261146497&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-1.500000000000&{}:{}&2.500000000000&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.350000000000&{}:{}&0.000000000000&{}:{}&-0.350000000000&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.865384615385&{}:{}&0.134615384615&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)