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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3b (123)

Malfatti circles

3b (123)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.656250000000&{}:{}&-0.976562500000&{}:{}&1.320312500000&, \\ P_{\mathbf{3b}}&{}\approx{}&-0.003211009174&{}:{}&1.254013761468&{}:{}&-0.250802752294&, \\ P^-_{\mathbf{3b}}&{}\approx{}&0.105857580398&{}:{}&0.885097626340&{}:{}&0.009044793262&, \\ P^+_{\mathbf{3b}}&{}\approx{}&-0.166189931350&{}:{}&1.805277459954&{}:{}&-0.639087528604&, \\ Q_{\mathbf{3b}}&{}\approx{}&0.280000000000&{}:{}&1.800000000000&{}:{}&-1.080000000000&, \\ I^\prime_{\mathbf{3b}}&{}\approx{}&-0.094594594595&{}:{}&2.280405405405&{}:{}&-1.185810810811&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{3b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{3b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{3b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{3b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{3b}}\) Radical center of the Malfatti circles
3b (123)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3b}}&{}\approx{}&2.160156250000&{}:{}&3.295898437500&{}:{}&-4.456054687500&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.194444444444&{}:{}&1.585648148148&{}:{}&-0.391203703704&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-1.750000000000&{}:{}&2.604166666667&{}:{}&0.145833333333&, \\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.350000000000&{}:{}&1.687500000000&{}:{}&-0.337500000000&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.024137931034&{}:{}&2.909482758621&{}:{}&-1.885344827586&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.004413619168&{}:{}&1.723675914250&{}:{}&-0.719262295082&, \\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.138157894737&{}:{}&1.126644736842&{}:{}&0.011513157895&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.382963988920&{}:{}&0.584314404432&{}:{}&0.032721606648&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.137016848365&{}:{}&1.145626858276&{}:{}&-0.282643706640&, \\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&2.500000000000&{}:{}&-1.500000000000&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.350000000000&{}:{}&0.000000000000&{}:{}&1.350000000000&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.134615384615&{}:{}&0.865384615385&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3b}}}{B^\prime_{\mathbf{3b}}}{C^\prime_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^*_{\mathbf{3b}}}{B^*_{\mathbf{3b}}}{C^*_{\mathbf{3b}}}\)
3b (123)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3b}}}}&{}\approx{}&-3.375000000000&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3b}}}}&{}\approx{}&-0.296296296296&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3b}}}}&{}\approx{}&-2.666666666667&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.656250000000&{}:{}&-0.976562500000&{}:{}&1.320312500000&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&2.160156250000&{}:{}&3.295898437500&{}:{}&-4.456054687500&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.194444444444&{}:{}&1.585648148148&{}:{}&-0.391203703704&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-1.750000000000&{}:{}&2.604166666667&{}:{}&0.145833333333&. \end{alignedat} \]
3b (123)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3b}}&{}\approx{}&-0.003211009174&{}:{}&1.254013761468&{}:{}&-0.250802752294&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.350000000000&{}:{}&1.687500000000&{}:{}&-0.337500000000&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.024137931034&{}:{}&2.909482758621&{}:{}&-1.885344827586&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.004413619168&{}:{}&1.723675914250&{}:{}&-0.719262295082&. \end{alignedat} \]
3b (123)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3b}}&{}\approx{}&0.105857580398&{}:{}&0.885097626340&{}:{}&0.009044793262&,\\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.138157894737&{}:{}&1.126644736842&{}:{}&0.011513157895&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.382963988920&{}:{}&0.584314404432&{}:{}&0.032721606648&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.137016848365&{}:{}&1.145626858276&{}:{}&-0.282643706640&. \end{alignedat} \]
3b (123)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3b}}&{}\approx{}&-0.166189931350&{}:{}&1.805277459954&{}:{}&-0.639087528604&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&2.160156250000&{}:{}&3.295898437500&{}:{}&-4.456054687500&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.194444444444&{}:{}&1.585648148148&{}:{}&-0.391203703704&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-1.750000000000&{}:{}&2.604166666667&{}:{}&0.145833333333&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.350000000000&{}:{}&1.687500000000&{}:{}&-0.337500000000&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.024137931034&{}:{}&2.909482758621&{}:{}&-1.885344827586&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.004413619168&{}:{}&1.723675914250&{}:{}&-0.719262295082&, \end{alignedat} \]
3b (123)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3b}}&{}\approx{}&0.280000000000&{}:{}&1.800000000000&{}:{}&-1.080000000000&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&2.500000000000&{}:{}&-1.500000000000&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.350000000000&{}:{}&0.000000000000&{}:{}&1.350000000000&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.134615384615&{}:{}&0.865384615385&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3b}}&{}\approx{}&-0.094594594595&{}:{}&2.280405405405&{}:{}&-1.185810810811&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.350000000000&{}:{}&1.687500000000&{}:{}&-0.337500000000&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.024137931034&{}:{}&2.909482758621&{}:{}&-1.885344827586&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.004413619168&{}:{}&1.723675914250&{}:{}&-0.719262295082&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&2.500000000000&{}:{}&-1.500000000000&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.350000000000&{}:{}&0.000000000000&{}:{}&1.350000000000&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.134615384615&{}:{}&0.865384615385&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)