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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4b(301)

Malfatti circles

4b (301)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.656250000000&{}:{}&-0.976562500000&{}:{}&1.320312500000&, \\ P_{\mathbf{4b}}&{}\approx{}&0.012741728798&{}:{}&-0.007961760252&{}:{}&0.995220031454&, \\ P^-_{\mathbf{4b}}&{}\approx{}&-0.008156422901&{}:{}&0.023493880803&{}:{}&0.984662542098&, \\ P^+_{\mathbf{4b}}&{}\approx{}&0.032365315112&{}:{}&-0.037498941157&{}:{}&1.005133626045&, \\ Q_{\mathbf{4b}}&{}\approx{}&-0.321100917431&{}:{}&0.082568807339&{}:{}&1.238532110092&, \\ I^\prime_{\mathbf{4b}}&{}\approx{}&0.079545454545&{}:{}&-0.076704545455&{}:{}&0.997159090909&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{4b}}\)
\(P^-_{\mathbf{4b}}\)
\(P^+_{\mathbf{4b}}\)
\(Q_{\mathbf{4b}}\)
\(I^\prime_{\mathbf{4b}}\)
4b (301)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4b}}&{}\approx{}&-2.712500000000&{}:{}&-10.546875000000&{}:{}&14.259375000000&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.243055555556&{}:{}&0.267939814815&{}:{}&0.489004629630&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.021875000000&{}:{}&-0.032552083333&{}:{}&1.010677083333&, \\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.040137614679&{}:{}&-0.007740825688&{}:{}&0.967603211009&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.013461538462&{}:{}&-0.064903846154&{}:{}&1.051442307692&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.145059681698&{}:{}&-0.090641578249&{}:{}&0.945581896552&, \\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.020701892744&{}:{}&0.022821372240&{}:{}&0.956476735016&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.008633036283&{}:{}&-0.033567393380&{}:{}&1.042200429663&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.140108471074&{}:{}&0.403570506198&{}:{}&0.736537964876&, \\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.062500000000&{}:{}&0.937500000000&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.350000000000&{}:{}&0.000000000000&{}:{}&1.350000000000&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.346153846154&{}:{}&-0.346153846154&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4b}}}{B^\prime_{\mathbf{4b}}}{C^\prime_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^*_{\mathbf{4b}}}{B^*_{\mathbf{4b}}}{C^*_{\mathbf{4b}}}\)
4b (301)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4b}}}}&{}\approx{}&10.800000000000&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4b}}}}&{}\approx{}&0.370370370370&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4b}}}}&{}\approx{}&0.033333333333&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.656250000000&{}:{}&-0.976562500000&{}:{}&1.320312500000&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&-2.712500000000&{}:{}&-10.546875000000&{}:{}&14.259375000000&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.243055555556&{}:{}&0.267939814815&{}:{}&0.489004629630&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.021875000000&{}:{}&-0.032552083333&{}:{}&1.010677083333&. \end{alignedat} \]
4b (301)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4b}}&{}\approx{}&0.012741728798&{}:{}&-0.007961760252&{}:{}&0.995220031454&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.040137614679&{}:{}&-0.007740825688&{}:{}&0.967603211009&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.013461538462&{}:{}&-0.064903846154&{}:{}&1.051442307692&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.145059681698&{}:{}&-0.090641578249&{}:{}&0.945581896552&. \end{alignedat} \]
4b (301)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4b}}&{}\approx{}&-0.008156422901&{}:{}&0.023493880803&{}:{}&0.984662542098&,\\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.020701892744&{}:{}&0.022821372240&{}:{}&0.956476735016&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.008633036283&{}:{}&-0.033567393380&{}:{}&1.042200429663&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.140108471074&{}:{}&0.403570506198&{}:{}&0.736537964876&. \end{alignedat} \]
4b (301)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4b}}&{}\approx{}&0.032365315112&{}:{}&-0.037498941157&{}:{}&1.005133626045&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&-2.712500000000&{}:{}&-10.546875000000&{}:{}&14.259375000000&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.243055555556&{}:{}&0.267939814815&{}:{}&0.489004629630&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.021875000000&{}:{}&-0.032552083333&{}:{}&1.010677083333&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.040137614679&{}:{}&-0.007740825688&{}:{}&0.967603211009&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.013461538462&{}:{}&-0.064903846154&{}:{}&1.051442307692&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.145059681698&{}:{}&-0.090641578249&{}:{}&0.945581896552&, \end{alignedat} \]
4b (301)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4b}}&{}\approx{}&-0.321100917431&{}:{}&0.082568807339&{}:{}&1.238532110092&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.062500000000&{}:{}&0.937500000000&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.350000000000&{}:{}&0.000000000000&{}:{}&1.350000000000&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.346153846154&{}:{}&-0.346153846154&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4b}}&{}\approx{}&0.079545454545&{}:{}&-0.076704545455&{}:{}&0.997159090909&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.040137614679&{}:{}&-0.007740825688&{}:{}&0.967603211009&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.013461538462&{}:{}&-0.064903846154&{}:{}&1.051442307692&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.145059681698&{}:{}&-0.090641578249&{}:{}&0.945581896552&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.062500000000&{}:{}&0.937500000000&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.350000000000&{}:{}&0.000000000000&{}:{}&1.350000000000&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.346153846154&{}:{}&-0.346153846154&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)