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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0c(110)

Malfatti circles

0c (110)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.100000000000&{}:{}&3.125000000000&{}:{}&-4.225000000000&, \\ P_{\mathbf{c}}&{}\approx{}&1.015297450425&{}:{}&1.223796033994&{}:{}&-1.239093484419&, \\ P^-_{\mathbf{c}}&{}\approx{}&0.844590163934&{}:{}&0.924590163934&{}:{}&-0.769180327869&, \\ P^+_{\mathbf{c}}&{}\approx{}&1.145137157107&{}:{}&1.451371571072&{}:{}&-1.596508728180&, \\ Q_{\mathbf{c}}&{}\approx{}&0.823529411765&{}:{}&0.705882352941&{}:{}&-0.529411764706&, \\ I^\prime_{\mathbf{c}}&{}\approx{}&1.294797687861&{}:{}&1.734104046243&{}:{}&-2.028901734104&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{c}}\)
\(P^-_{\mathbf{c}}\)
\(P^+_{\mathbf{c}}\)
\(Q_{\mathbf{c}}\)
\(I^\prime_{\mathbf{c}}\)
0c (110)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{c}}&{}\approx{}&1.257812500000&{}:{}&0.732421875000&{}:{}&-0.990234375000&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.560000000000&{}:{}&1.566666666667&{}:{}&-1.126666666667&,\\C^\prime_{\mathbf{c}}&{}\approx{}&6.222222222222&{}:{}&9.259259259259&{}:{}&-14.481481481481&, \\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.027522935780&{}:{}&2.201834862385&{}:{}&-2.229357798165&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.621719457014&{}:{}&1.357466063348&{}:{}&-1.979185520362&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.931392931393&{}:{}&1.122661122661&{}:{}&-1.054054054054&, \\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.680161943320&{}:{}&1.902834008097&{}:{}&-1.582995951417&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.489017341040&{}:{}&0.867052023121&{}:{}&-1.356069364162&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.764845605701&{}:{}&0.837292161520&{}:{}&-0.602137767221&, \\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&4.000000000000&{}:{}&-3.000000000000&,\\B^*_{\mathbf{c}}&{}\approx{}&2.800000000000&{}:{}&0.000000000000&{}:{}&-1.800000000000&,\\C^*_{\mathbf{c}}&{}\approx{}&0.538461538462&{}:{}&0.461538461538&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{c}}}{B^\prime_{\mathbf{c}}}{C^\prime_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{c}}}{B^{\prime\prime}_{\mathbf{c}}}{C^{\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{c}}}{B^{\prime\prime\prime}_{\mathbf{c}}}{C^{\prime\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^*_{\mathbf{c}}}{B^*_{\mathbf{c}}}{C^*_{\mathbf{c}}}\)
0c (110)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{c}}}}&{}\approx{}&0.234375000000&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{c}}}}&{}\approx{}&0.266666666667&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{c}}}}&{}\approx{}&2.962962962963&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.100000000000&{}:{}&3.125000000000&{}:{}&-4.225000000000&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.257812500000&{}:{}&0.732421875000&{}:{}&-0.990234375000&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.560000000000&{}:{}&1.566666666667&{}:{}&-1.126666666667&,\\C^\prime_{\mathbf{c}}&{}\approx{}&6.222222222222&{}:{}&9.259259259259&{}:{}&-14.481481481481&. \end{alignedat} \]
0c (110)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{c}}&{}\approx{}&1.015297450425&{}:{}&1.223796033994&{}:{}&-1.239093484419&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.027522935780&{}:{}&2.201834862385&{}:{}&-2.229357798165&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.621719457014&{}:{}&1.357466063348&{}:{}&-1.979185520362&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.931392931393&{}:{}&1.122661122661&{}:{}&-1.054054054054&. \end{alignedat} \]
0c (110)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{c}}&{}\approx{}&0.844590163934&{}:{}&0.924590163934&{}:{}&-0.769180327869&,\\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.680161943320&{}:{}&1.902834008097&{}:{}&-1.582995951417&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.489017341040&{}:{}&0.867052023121&{}:{}&-1.356069364162&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.764845605701&{}:{}&0.837292161520&{}:{}&-0.602137767221&. \end{alignedat} \]
0c (110)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{c}}&{}\approx{}&1.145137157107&{}:{}&1.451371571072&{}:{}&-1.596508728180&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.257812500000&{}:{}&0.732421875000&{}:{}&-0.990234375000&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.560000000000&{}:{}&1.566666666667&{}:{}&-1.126666666667&,\\C^\prime_{\mathbf{c}}&{}\approx{}&6.222222222222&{}:{}&9.259259259259&{}:{}&-14.481481481481&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.027522935780&{}:{}&2.201834862385&{}:{}&-2.229357798165&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.621719457014&{}:{}&1.357466063348&{}:{}&-1.979185520362&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.931392931393&{}:{}&1.122661122661&{}:{}&-1.054054054054&, \end{alignedat} \]
0c (110)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{c}}&{}\approx{}&0.823529411765&{}:{}&0.705882352941&{}:{}&-0.529411764706&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&4.000000000000&{}:{}&-3.000000000000&,\\B^*_{\mathbf{c}}&{}\approx{}&2.800000000000&{}:{}&0.000000000000&{}:{}&-1.800000000000&,\\C^*_{\mathbf{c}}&{}\approx{}&0.538461538462&{}:{}&0.461538461538&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{c}}&{}\approx{}&1.294797687861&{}:{}&1.734104046243&{}:{}&-2.028901734104&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.027522935780&{}:{}&2.201834862385&{}:{}&-2.229357798165&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.621719457014&{}:{}&1.357466063348&{}:{}&-1.979185520362&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.931392931393&{}:{}&1.122661122661&{}:{}&-1.054054054054&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&4.000000000000&{}:{}&-3.000000000000&,\\B^*_{\mathbf{c}}&{}\approx{}&2.800000000000&{}:{}&0.000000000000&{}:{}&-1.800000000000&,\\C^*_{\mathbf{c}}&{}\approx{}&0.538461538462&{}:{}&0.461538461538&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)