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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0b(101)

Malfatti circles

0b (101)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.656250000000&{}:{}&-0.976562500000&{}:{}&1.320312500000&, \\ P_{\mathbf{b}}&{}\approx{}&0.746587030717&{}:{}&-0.466510238908&{}:{}&0.719923208191&, \\ P^-_{\mathbf{b}}&{}\approx{}&0.769849785408&{}:{}&-0.335166309013&{}:{}&0.565316523605&, \\ P^+_{\mathbf{b}}&{}\approx{}&0.731232294618&{}:{}&-0.553204674221&{}:{}&0.821972379603&, \\ Q_{\mathbf{b}}&{}\approx{}&0.853658536585&{}:{}&-0.219512195122&{}:{}&0.365853658537&, \\ I^\prime_{\mathbf{b}}&{}\approx{}&0.700000000000&{}:{}&-0.675000000000&{}:{}&0.975000000000&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{b}}\) Radical center of the Malfatti circles
0b (101)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{b}}&{}\approx{}&0.896875000000&{}:{}&-0.292968750000&{}:{}&0.396093750000&,\\B^\prime_{\mathbf{b}}&{}\approx{}&0.972222222222&{}:{}&-1.928240740741&{}:{}&1.956018518519&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.350000000000&{}:{}&-0.520833333333&{}:{}&1.170833333333&, \\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.514705882353&{}:{}&-0.893382352941&{}:{}&1.378676470588&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.700000000000&{}:{}&-0.375000000000&{}:{}&0.675000000000&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.909563409563&{}:{}&-0.568347193347&{}:{}&0.658783783784&, \\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.423387096774&{}:{}&-0.839717741935&{}:{}&1.416330645161&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.710396039604&{}:{}&-0.232054455446&{}:{}&0.521658415842&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.993767313019&{}:{}&-0.432652354571&{}:{}&0.438885041551&, \\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-1.500000000000&{}:{}&2.500000000000&,\\B^*_{\mathbf{b}}&{}\approx{}&0.700000000000&{}:{}&0.000000000000&{}:{}&0.300000000000&,\\C^*_{\mathbf{b}}&{}\approx{}&1.346153846154&{}:{}&-0.346153846154&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{b}}}{B^\prime_{\mathbf{b}}}{C^\prime_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{b}}}{B^{\prime\prime}_{\mathbf{b}}}{C^{\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{b}}}{B^{\prime\prime\prime}_{\mathbf{b}}}{C^{\prime\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^*_{\mathbf{b}}}{B^*_{\mathbf{b}}}{C^*_{\mathbf{b}}}\)
0b (101)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{b}}}}&{}\approx{}&0.300000000000&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{b}}}}&{}\approx{}&1.481481481481&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{b}}}}&{}\approx{}&0.533333333333&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.656250000000&{}:{}&-0.976562500000&{}:{}&1.320312500000&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.896875000000&{}:{}&-0.292968750000&{}:{}&0.396093750000&,\\B^\prime_{\mathbf{b}}&{}\approx{}&0.972222222222&{}:{}&-1.928240740741&{}:{}&1.956018518519&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.350000000000&{}:{}&-0.520833333333&{}:{}&1.170833333333&. \end{alignedat} \]
0b (101)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{b}}&{}\approx{}&0.746587030717&{}:{}&-0.466510238908&{}:{}&0.719923208191&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.514705882353&{}:{}&-0.893382352941&{}:{}&1.378676470588&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.700000000000&{}:{}&-0.375000000000&{}:{}&0.675000000000&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.909563409563&{}:{}&-0.568347193347&{}:{}&0.658783783784&. \end{alignedat} \]
0b (101)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{b}}&{}\approx{}&0.769849785408&{}:{}&-0.335166309013&{}:{}&0.565316523605&,\\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.423387096774&{}:{}&-0.839717741935&{}:{}&1.416330645161&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.710396039604&{}:{}&-0.232054455446&{}:{}&0.521658415842&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.993767313019&{}:{}&-0.432652354571&{}:{}&0.438885041551&. \end{alignedat} \]
0b (101)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{b}}&{}\approx{}&0.731232294618&{}:{}&-0.553204674221&{}:{}&0.821972379603&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.896875000000&{}:{}&-0.292968750000&{}:{}&0.396093750000&,\\B^\prime_{\mathbf{b}}&{}\approx{}&0.972222222222&{}:{}&-1.928240740741&{}:{}&1.956018518519&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.350000000000&{}:{}&-0.520833333333&{}:{}&1.170833333333&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.514705882353&{}:{}&-0.893382352941&{}:{}&1.378676470588&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.700000000000&{}:{}&-0.375000000000&{}:{}&0.675000000000&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.909563409563&{}:{}&-0.568347193347&{}:{}&0.658783783784&, \end{alignedat} \]
0b (101)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{b}}&{}\approx{}&0.853658536585&{}:{}&-0.219512195122&{}:{}&0.365853658537&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-1.500000000000&{}:{}&2.500000000000&,\\B^*_{\mathbf{b}}&{}\approx{}&0.700000000000&{}:{}&0.000000000000&{}:{}&0.300000000000&,\\C^*_{\mathbf{b}}&{}\approx{}&1.346153846154&{}:{}&-0.346153846154&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{b}}&{}\approx{}&0.700000000000&{}:{}&-0.675000000000&{}:{}&0.975000000000&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.514705882353&{}:{}&-0.893382352941&{}:{}&1.378676470588&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.700000000000&{}:{}&-0.375000000000&{}:{}&0.675000000000&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.909563409563&{}:{}&-0.568347193347&{}:{}&0.658783783784&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-1.500000000000&{}:{}&2.500000000000&,\\B^*_{\mathbf{b}}&{}\approx{}&0.700000000000&{}:{}&0.000000000000&{}:{}&0.300000000000&,\\C^*_{\mathbf{b}}&{}\approx{}&1.346153846154&{}:{}&-0.346153846154&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)