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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5a(213)

Malfatti circles

5a (213)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.400000000000&{}:{}&0.595238095238&{}:{}&0.804761904762&, \\ P_{\mathbf{5a}}&{}\approx{}&1.050810810811&{}:{}&-0.000772200772&{}:{}&-0.050038610039&, \\ P^-_{\mathbf{5a}}&{}\approx{}&0.866037735849&{}:{}&0.075134770889&{}:{}&0.058827493261&, \\ P^+_{\mathbf{5a}}&{}\approx{}&1.298734177215&{}:{}&-0.102622061483&{}:{}&-0.196112115732&, \\ Q_{\mathbf{5a}}&{}\approx{}&1.411764705882&{}:{}&0.117647058824&{}:{}&-0.529411764706&, \\ I^\prime_{\mathbf{5a}}&{}\approx{}&1.521126760563&{}:{}&-0.050301810865&{}:{}&-0.470824949698&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{5a}}\)
\(P^-_{\mathbf{5a}}\)
\(P^+_{\mathbf{5a}}\)
\(Q_{\mathbf{5a}}\)
\(I^\prime_{\mathbf{5a}}\)
5a (213)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5a}}&{}\approx{}&1.259259259259&{}:{}&-0.110229276896&{}:{}&-0.149029982363&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&2.160000000000&{}:{}&3.185714285714&{}:{}&-4.345714285714&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.500000000000&{}:{}&-2.232142857143&{}:{}&1.732142857143&, \\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.770491803279&{}:{}&-0.011709601874&{}:{}&-0.758782201405&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.270588235294&{}:{}&-0.210084033613&{}:{}&-0.060504201681&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.289124668435&{}:{}&-0.000947328534&{}:{}&-0.288177339901&, \\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.275510204082&{}:{}&0.406341107872&{}:{}&0.318148688047&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.020000000000&{}:{}&-0.089285714286&{}:{}&0.069285714286&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.032620922385&{}:{}&0.089587015909&{}:{}&-0.122207938293&, \\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.285714285714&{}:{}&1.285714285714&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.600000000000&{}:{}&0.000000000000&{}:{}&-0.600000000000&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.923076923077&{}:{}&0.076923076923&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5a}}}{B^\prime_{\mathbf{5a}}}{C^\prime_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^*_{\mathbf{5a}}}{B^*_{\mathbf{5a}}}{C^*_{\mathbf{5a}}}\)
5a (213)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5a}}}}&{}\approx{}&-0.185185185185&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5a}}}}&{}\approx{}&-5.400000000000&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5a}}}}&{}\approx{}&-3.750000000000&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.400000000000&{}:{}&0.595238095238&{}:{}&0.804761904762&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.259259259259&{}:{}&-0.110229276896&{}:{}&-0.149029982363&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&2.160000000000&{}:{}&3.185714285714&{}:{}&-4.345714285714&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.500000000000&{}:{}&-2.232142857143&{}:{}&1.732142857143&. \end{alignedat} \]
5a (213)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5a}}&{}\approx{}&1.050810810811&{}:{}&-0.000772200772&{}:{}&-0.050038610039&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.770491803279&{}:{}&-0.011709601874&{}:{}&-0.758782201405&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.270588235294&{}:{}&-0.210084033613&{}:{}&-0.060504201681&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.289124668435&{}:{}&-0.000947328534&{}:{}&-0.288177339901&. \end{alignedat} \]
5a (213)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5a}}&{}\approx{}&0.866037735849&{}:{}&0.075134770889&{}:{}&0.058827493261&,\\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.275510204082&{}:{}&0.406341107872&{}:{}&0.318148688047&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.020000000000&{}:{}&-0.089285714286&{}:{}&0.069285714286&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.032620922385&{}:{}&0.089587015909&{}:{}&-0.122207938293&. \end{alignedat} \]
5a (213)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5a}}&{}\approx{}&1.298734177215&{}:{}&-0.102622061483&{}:{}&-0.196112115732&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.259259259259&{}:{}&-0.110229276896&{}:{}&-0.149029982363&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&2.160000000000&{}:{}&3.185714285714&{}:{}&-4.345714285714&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.500000000000&{}:{}&-2.232142857143&{}:{}&1.732142857143&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.770491803279&{}:{}&-0.011709601874&{}:{}&-0.758782201405&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.270588235294&{}:{}&-0.210084033613&{}:{}&-0.060504201681&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.289124668435&{}:{}&-0.000947328534&{}:{}&-0.288177339901&, \end{alignedat} \]
5a (213)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5a}}&{}\approx{}&1.411764705882&{}:{}&0.117647058824&{}:{}&-0.529411764706&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.285714285714&{}:{}&1.285714285714&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.600000000000&{}:{}&0.000000000000&{}:{}&-0.600000000000&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.923076923077&{}:{}&0.076923076923&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5a}}&{}\approx{}&1.521126760563&{}:{}&-0.050301810865&{}:{}&-0.470824949698&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.770491803279&{}:{}&-0.011709601874&{}:{}&-0.758782201405&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.270588235294&{}:{}&-0.210084033613&{}:{}&-0.060504201681&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.289124668435&{}:{}&-0.000947328534&{}:{}&-0.288177339901&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.285714285714&{}:{}&1.285714285714&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.600000000000&{}:{}&0.000000000000&{}:{}&-0.600000000000&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.923076923077&{}:{}&0.076923076923&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)