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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2b (121)

Malfatti circles

2b (121)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.656250000000&{}:{}&-0.976562500000&{}:{}&1.320312500000&, \\ P_{\mathbf{2b}}&{}\approx{}&-0.008823529412&{}:{}&1.261029411765&{}:{}&-0.252205882353&, \\ P^-_{\mathbf{2b}}&{}\approx{}&0.155973451327&{}:{}&0.706581858407&{}:{}&0.137444690265&, \\ P^+_{\mathbf{2b}}&{}\approx{}&-0.335526315789&{}:{}&2.360197368421&{}:{}&-1.024671052632&, \\ Q_{\mathbf{2b}}&{}\approx{}&-1.800000000000&{}:{}&7.000000000000&{}:{}&-4.200000000000&, \\ I^\prime_{\mathbf{2b}}&{}\approx{}&-0.166666666667&{}:{}&2.430555555556&{}:{}&-1.263888888889&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{2b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{2b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{2b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{2b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{2b}}\) Radical center of the Malfatti circles
2b (121)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2b}}&{}\approx{}&2.203125000000&{}:{}&3.417968750000&{}:{}&-4.621093750000&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.750000000000&{}:{}&3.258928571429&{}:{}&-1.508928571429&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.750000000000&{}:{}&1.116071428571&{}:{}&0.633928571429&, \\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.750000000000&{}:{}&2.187500000000&{}:{}&-0.437500000000&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.023076923077&{}:{}&1.682692307692&{}:{}&-0.659615384615&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.023076923077&{}:{}&3.298076923077&{}:{}&-2.275000000000&, \\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.238636363636&{}:{}&1.036931818182&{}:{}&0.201704545455&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.291322314050&{}:{}&0.451962809917&{}:{}&0.256714876033&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.291322314050&{}:{}&1.319731404959&{}:{}&-0.611053719008&, \\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&2.500000000000&{}:{}&-1.500000000000&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.300000000000&{}:{}&0.000000000000&{}:{}&0.700000000000&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.346153846154&{}:{}&1.346153846154&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2b}}}{B^\prime_{\mathbf{2b}}}{C^\prime_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^*_{\mathbf{2b}}}{B^*_{\mathbf{2b}}}{C^*_{\mathbf{2b}}}\)
2b (121)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2b}}}}&{}\approx{}&-3.500000000000&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2b}}}}&{}\approx{}&-1.142857142857&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2b}}}}&{}\approx{}&-1.142857142857&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.656250000000&{}:{}&-0.976562500000&{}:{}&1.320312500000&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&2.203125000000&{}:{}&3.417968750000&{}:{}&-4.621093750000&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.750000000000&{}:{}&3.258928571429&{}:{}&-1.508928571429&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.750000000000&{}:{}&1.116071428571&{}:{}&0.633928571429&. \end{alignedat} \]
2b (121)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2b}}&{}\approx{}&-0.008823529412&{}:{}&1.261029411765&{}:{}&-0.252205882353&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.750000000000&{}:{}&2.187500000000&{}:{}&-0.437500000000&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.023076923077&{}:{}&1.682692307692&{}:{}&-0.659615384615&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.023076923077&{}:{}&3.298076923077&{}:{}&-2.275000000000&. \end{alignedat} \]
2b (121)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2b}}&{}\approx{}&0.155973451327&{}:{}&0.706581858407&{}:{}&0.137444690265&,\\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.238636363636&{}:{}&1.036931818182&{}:{}&0.201704545455&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.291322314050&{}:{}&0.451962809917&{}:{}&0.256714876033&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.291322314050&{}:{}&1.319731404959&{}:{}&-0.611053719008&. \end{alignedat} \]
2b (121)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2b}}&{}\approx{}&-0.335526315789&{}:{}&2.360197368421&{}:{}&-1.024671052632&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&2.203125000000&{}:{}&3.417968750000&{}:{}&-4.621093750000&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.750000000000&{}:{}&3.258928571429&{}:{}&-1.508928571429&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.750000000000&{}:{}&1.116071428571&{}:{}&0.633928571429&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.750000000000&{}:{}&2.187500000000&{}:{}&-0.437500000000&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.023076923077&{}:{}&1.682692307692&{}:{}&-0.659615384615&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.023076923077&{}:{}&3.298076923077&{}:{}&-2.275000000000&, \end{alignedat} \]
2b (121)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2b}}&{}\approx{}&-1.800000000000&{}:{}&7.000000000000&{}:{}&-4.200000000000&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&2.500000000000&{}:{}&-1.500000000000&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.300000000000&{}:{}&0.000000000000&{}:{}&0.700000000000&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.346153846154&{}:{}&1.346153846154&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2b}}&{}\approx{}&-0.166666666667&{}:{}&2.430555555556&{}:{}&-1.263888888889&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.750000000000&{}:{}&2.187500000000&{}:{}&-0.437500000000&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.023076923077&{}:{}&1.682692307692&{}:{}&-0.659615384615&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.023076923077&{}:{}&3.298076923077&{}:{}&-2.275000000000&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&2.500000000000&{}:{}&-1.500000000000&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.300000000000&{}:{}&0.000000000000&{}:{}&0.700000000000&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.346153846154&{}:{}&1.346153846154&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)