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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4a(211)

Malfatti circles

4a (211)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.400000000000&{}:{}&0.595238095238&{}:{}&0.804761904762&, \\ P_{\mathbf{4a}}&{}\approx{}&1.054117647059&{}:{}&-0.003921568627&{}:{}&-0.050196078431&, \\ P^-_{\mathbf{4a}}&{}\approx{}&0.693805309735&{}:{}&0.144542772861&{}:{}&0.161651917404&, \\ P^+_{\mathbf{4a}}&{}\approx{}&1.768421052632&{}:{}&-0.298245614035&{}:{}&-0.470175438596&, \\ Q_{\mathbf{4a}}&{}\approx{}&2.285714285714&{}:{}&-0.428571428571&{}:{}&-0.857142857143&, \\ I^\prime_{\mathbf{4a}}&{}\approx{}&1.623188405797&{}:{}&-0.120772946860&{}:{}&-0.502415458937&, \end{alignedat} \]
\(I_{\mathbf{a}}\) Incenter
\(P_{\mathbf{4a}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{4a}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{4a}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{4a}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{4a}}\) Radical center of the Malfatti circles
4a (211)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4a}}&{}\approx{}&2.000000000000&{}:{}&-0.425170068027&{}:{}&-0.574829931973&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&2.240000000000&{}:{}&3.266666666667&{}:{}&-4.506666666667&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.875000000000&{}:{}&-1.302083333333&{}:{}&1.427083333333&, \\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.258426966292&{}:{}&-0.018726591760&{}:{}&-0.239700374532&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.723076923077&{}:{}&-0.641025641026&{}:{}&-0.082051282051&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.027149321267&{}:{}&-0.007541478130&{}:{}&-1.019607843137&, \\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.244541484716&{}:{}&0.356622998544&{}:{}&0.398835516739&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.980000000000&{}:{}&-0.208333333333&{}:{}&0.228333333333&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.085872576177&{}:{}&0.226223453370&{}:{}&-0.312096029548&, \\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.333333333333&{}:{}&0.666666666667&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.600000000000&{}:{}&0.000000000000&{}:{}&-0.600000000000&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.230769230769&{}:{}&-0.230769230769&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4a}}}{B^\prime_{\mathbf{4a}}}{C^\prime_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^*_{\mathbf{4a}}}{B^*_{\mathbf{4a}}}{C^*_{\mathbf{4a}}}\)
4a (211)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4a}}}}&{}\approx{}&-0.714285714286&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4a}}}}&{}\approx{}&-5.600000000000&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4a}}}}&{}\approx{}&-2.187500000000&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.400000000000&{}:{}&0.595238095238&{}:{}&0.804761904762&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&2.000000000000&{}:{}&-0.425170068027&{}:{}&-0.574829931973&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&2.240000000000&{}:{}&3.266666666667&{}:{}&-4.506666666667&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.875000000000&{}:{}&-1.302083333333&{}:{}&1.427083333333&. \end{alignedat} \]
4a (211)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4a}}&{}\approx{}&1.054117647059&{}:{}&-0.003921568627&{}:{}&-0.050196078431&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.258426966292&{}:{}&-0.018726591760&{}:{}&-0.239700374532&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.723076923077&{}:{}&-0.641025641026&{}:{}&-0.082051282051&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.027149321267&{}:{}&-0.007541478130&{}:{}&-1.019607843137&. \end{alignedat} \]
4a (211)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4a}}&{}\approx{}&0.693805309735&{}:{}&0.144542772861&{}:{}&0.161651917404&,\\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.244541484716&{}:{}&0.356622998544&{}:{}&0.398835516739&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.980000000000&{}:{}&-0.208333333333&{}:{}&0.228333333333&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.085872576177&{}:{}&0.226223453370&{}:{}&-0.312096029548&. \end{alignedat} \]
4a (211)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4a}}&{}\approx{}&1.768421052632&{}:{}&-0.298245614035&{}:{}&-0.470175438596&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&2.000000000000&{}:{}&-0.425170068027&{}:{}&-0.574829931973&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&2.240000000000&{}:{}&3.266666666667&{}:{}&-4.506666666667&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.875000000000&{}:{}&-1.302083333333&{}:{}&1.427083333333&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.258426966292&{}:{}&-0.018726591760&{}:{}&-0.239700374532&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.723076923077&{}:{}&-0.641025641026&{}:{}&-0.082051282051&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.027149321267&{}:{}&-0.007541478130&{}:{}&-1.019607843137&, \end{alignedat} \]
4a (211)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4a}}&{}\approx{}&2.285714285714&{}:{}&-0.428571428571&{}:{}&-0.857142857143&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.333333333333&{}:{}&0.666666666667&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.600000000000&{}:{}&0.000000000000&{}:{}&-0.600000000000&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.230769230769&{}:{}&-0.230769230769&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4a}}&{}\approx{}&1.623188405797&{}:{}&-0.120772946860&{}:{}&-0.502415458937&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.258426966292&{}:{}&-0.018726591760&{}:{}&-0.239700374532&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.723076923077&{}:{}&-0.641025641026&{}:{}&-0.082051282051&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.027149321267&{}:{}&-0.007541478130&{}:{}&-1.019607843137&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.333333333333&{}:{}&0.666666666667&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.600000000000&{}:{}&0.000000000000&{}:{}&-0.600000000000&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.230769230769&{}:{}&-0.230769230769&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)