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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7a (233)

Malfatti circles

7a (233)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.400000000000&{}:{}&0.595238095238&{}:{}&0.804761904762&, \\ P_{\mathbf{7a}}&{}\approx{}&1.001324503311&{}:{}&-0.000735835173&{}:{}&-0.000588668138&, \\ P^-_{\mathbf{7a}}&{}\approx{}&0.980000000000&{}:{}&0.008333333333&{}:{}&0.011666666667&, \\ P^+_{\mathbf{7a}}&{}\approx{}&1.023318385650&{}:{}&-0.010089686099&{}:{}&-0.013228699552&, \\ Q_{\mathbf{7a}}&{}\approx{}&0.888888888889&{}:{}&0.074074074074&{}:{}&0.037037037037&, \\ I^\prime_{\mathbf{7a}}&{}\approx{}&1.072340425532&{}:{}&-0.035460992908&{}:{}&-0.036879432624&, \end{alignedat} \]
\(I_{\mathbf{a}}\) Incenter
\(P_{\mathbf{7a}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{7a}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{7a}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{7a}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{7a}}\) Radical center of the Malfatti circles
7a (233)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7a}}&{}\approx{}&1.022222222222&{}:{}&-0.009448223734&{}:{}&-0.012773998488&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&2.800000000000&{}:{}&3.833333333333&{}:{}&-5.633333333333&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.700000000000&{}:{}&-1.041666666667&{}:{}&1.341666666667&, \\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.120000000000&{}:{}&-0.066666666667&{}:{}&-0.053333333333&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.019325842697&{}:{}&-0.018726591760&{}:{}&-0.000599250936&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.026244343891&{}:{}&-0.000754147813&{}:{}&-0.025490196078&, \\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.233333333333&{}:{}&0.319444444444&{}:{}&0.447222222222&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.997345132743&{}:{}&-0.009218289086&{}:{}&0.011873156342&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.004008908686&{}:{}&0.008537490720&{}:{}&-0.012546399406&, \\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.666666666667&{}:{}&0.333333333333&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.960000000000&{}:{}&0.000000000000&{}:{}&0.040000000000&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.923076923077&{}:{}&0.076923076923&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7a}}}{B^\prime_{\mathbf{7a}}}{C^\prime_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^*_{\mathbf{7a}}}{B^*_{\mathbf{7a}}}{C^*_{\mathbf{7a}}}\)
7a (233)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7a}}}}&{}\approx{}&-0.015873015873&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7a}}}}&{}\approx{}&-7.000000000000&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7a}}}}&{}\approx{}&-1.750000000000&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.400000000000&{}:{}&0.595238095238&{}:{}&0.804761904762&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.022222222222&{}:{}&-0.009448223734&{}:{}&-0.012773998488&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&2.800000000000&{}:{}&3.833333333333&{}:{}&-5.633333333333&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.700000000000&{}:{}&-1.041666666667&{}:{}&1.341666666667&. \end{alignedat} \]
7a (233)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7a}}&{}\approx{}&1.001324503311&{}:{}&-0.000735835173&{}:{}&-0.000588668138&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.120000000000&{}:{}&-0.066666666667&{}:{}&-0.053333333333&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.019325842697&{}:{}&-0.018726591760&{}:{}&-0.000599250936&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.026244343891&{}:{}&-0.000754147813&{}:{}&-0.025490196078&. \end{alignedat} \]
7a (233)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7a}}&{}\approx{}&0.980000000000&{}:{}&0.008333333333&{}:{}&0.011666666667&,\\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.233333333333&{}:{}&0.319444444444&{}:{}&0.447222222222&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.997345132743&{}:{}&-0.009218289086&{}:{}&0.011873156342&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.004008908686&{}:{}&0.008537490720&{}:{}&-0.012546399406&. \end{alignedat} \]
7a (233)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7a}}&{}\approx{}&1.023318385650&{}:{}&-0.010089686099&{}:{}&-0.013228699552&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.022222222222&{}:{}&-0.009448223734&{}:{}&-0.012773998488&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&2.800000000000&{}:{}&3.833333333333&{}:{}&-5.633333333333&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.700000000000&{}:{}&-1.041666666667&{}:{}&1.341666666667&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.120000000000&{}:{}&-0.066666666667&{}:{}&-0.053333333333&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.019325842697&{}:{}&-0.018726591760&{}:{}&-0.000599250936&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.026244343891&{}:{}&-0.000754147813&{}:{}&-0.025490196078&, \end{alignedat} \]
7a (233)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7a}}&{}\approx{}&0.888888888889&{}:{}&0.074074074074&{}:{}&0.037037037037&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.666666666667&{}:{}&0.333333333333&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.960000000000&{}:{}&0.000000000000&{}:{}&0.040000000000&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.923076923077&{}:{}&0.076923076923&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7a}}&{}\approx{}&1.072340425532&{}:{}&-0.035460992908&{}:{}&-0.036879432624&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.120000000000&{}:{}&-0.066666666667&{}:{}&-0.053333333333&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.019325842697&{}:{}&-0.018726591760&{}:{}&-0.000599250936&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.026244343891&{}:{}&-0.000754147813&{}:{}&-0.025490196078&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.666666666667&{}:{}&0.333333333333&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.960000000000&{}:{}&0.000000000000&{}:{}&0.040000000000&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.923076923077&{}:{}&0.076923076923&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)