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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4c(310)

Malfatti circles

4c (310)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.100000000000&{}:{}&3.125000000000&{}:{}&-4.225000000000&, \\ P_{\mathbf{4c}}&{}\approx{}&0.170018975332&{}:{}&1.037476280835&{}:{}&-0.207495256167&, \\ P^-_{\mathbf{4c}}&{}\approx{}&0.065800000000&{}:{}&0.924750000000&{}:{}&0.009450000000&, \\ P^+_{\mathbf{4c}}&{}\approx{}&0.264079422383&{}:{}&1.139214801444&{}:{}&-0.403294223827&, \\ Q_{\mathbf{4c}}&{}\approx{}&-0.636363636364&{}:{}&1.227272727273&{}:{}&0.409090909091&, \\ I^\prime_{\mathbf{4c}}&{}\approx{}&0.408759124088&{}:{}&1.231751824818&{}:{}&-0.640510948905&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{4c}}\)
\(P^-_{\mathbf{4c}}\)
\(P^+_{\mathbf{4c}}\)
\(Q_{\mathbf{4c}}\)
\(I^\prime_{\mathbf{4c}}\)
4c (310)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4c}}&{}\approx{}&10.281250000000&{}:{}&26.367187500000&{}:{}&-35.648437500000&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.140000000000&{}:{}&1.141666666667&{}:{}&-0.281666666667&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.388888888889&{}:{}&0.578703703704&{}:{}&0.032407407407&, \\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.205882352941&{}:{}&0.992647058824&{}:{}&-0.198529411765&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.601342281879&{}:{}&1.132550335570&{}:{}&-0.733892617450&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.219500244978&{}:{}&1.339416952474&{}:{}&-0.558917197452&, \\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.108247422680&{}:{}&0.882731958763&{}:{}&0.009020618557&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.269672131148&{}:{}&0.691598360656&{}:{}&0.038729508197&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.086306400839&{}:{}&1.212945960126&{}:{}&-0.299252360965&, \\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.750000000000&{}:{}&0.250000000000&,\\B^*_{\mathbf{4c}}&{}\approx{}&2.800000000000&{}:{}&0.000000000000&{}:{}&-1.800000000000&,\\C^*_{\mathbf{4c}}&{}\approx{}&-1.076923076923&{}:{}&2.076923076923&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4c}}}{B^\prime_{\mathbf{4c}}}{C^\prime_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^*_{\mathbf{4c}}}{B^*_{\mathbf{4c}}}{C^*_{\mathbf{4c}}}\)
4c (310)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4c}}}}&{}\approx{}&8.437500000000&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4c}}}}&{}\approx{}&0.066666666667&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4c}}}}&{}\approx{}&0.185185185185&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.100000000000&{}:{}&3.125000000000&{}:{}&-4.225000000000&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&10.281250000000&{}:{}&26.367187500000&{}:{}&-35.648437500000&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.140000000000&{}:{}&1.141666666667&{}:{}&-0.281666666667&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.388888888889&{}:{}&0.578703703704&{}:{}&0.032407407407&. \end{alignedat} \]
4c (310)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4c}}&{}\approx{}&0.170018975332&{}:{}&1.037476280835&{}:{}&-0.207495256167&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.205882352941&{}:{}&0.992647058824&{}:{}&-0.198529411765&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.601342281879&{}:{}&1.132550335570&{}:{}&-0.733892617450&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.219500244978&{}:{}&1.339416952474&{}:{}&-0.558917197452&. \end{alignedat} \]
4c (310)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4c}}&{}\approx{}&0.065800000000&{}:{}&0.924750000000&{}:{}&0.009450000000&,\\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.108247422680&{}:{}&0.882731958763&{}:{}&0.009020618557&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.269672131148&{}:{}&0.691598360656&{}:{}&0.038729508197&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.086306400839&{}:{}&1.212945960126&{}:{}&-0.299252360965&. \end{alignedat} \]
4c (310)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4c}}&{}\approx{}&0.264079422383&{}:{}&1.139214801444&{}:{}&-0.403294223827&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&10.281250000000&{}:{}&26.367187500000&{}:{}&-35.648437500000&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.140000000000&{}:{}&1.141666666667&{}:{}&-0.281666666667&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.388888888889&{}:{}&0.578703703704&{}:{}&0.032407407407&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.205882352941&{}:{}&0.992647058824&{}:{}&-0.198529411765&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.601342281879&{}:{}&1.132550335570&{}:{}&-0.733892617450&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.219500244978&{}:{}&1.339416952474&{}:{}&-0.558917197452&, \end{alignedat} \]
4c (310)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4c}}&{}\approx{}&-0.636363636364&{}:{}&1.227272727273&{}:{}&0.409090909091&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.750000000000&{}:{}&0.250000000000&,\\B^*_{\mathbf{4c}}&{}\approx{}&2.800000000000&{}:{}&0.000000000000&{}:{}&-1.800000000000&,\\C^*_{\mathbf{4c}}&{}\approx{}&-1.076923076923&{}:{}&2.076923076923&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4c}}&{}\approx{}&0.408759124088&{}:{}&1.231751824818&{}:{}&-0.640510948905&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.205882352941&{}:{}&0.992647058824&{}:{}&-0.198529411765&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.601342281879&{}:{}&1.132550335570&{}:{}&-0.733892617450&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.219500244978&{}:{}&1.339416952474&{}:{}&-0.558917197452&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.750000000000&{}:{}&0.250000000000&,\\B^*_{\mathbf{4c}}&{}\approx{}&2.800000000000&{}:{}&0.000000000000&{}:{}&-1.800000000000&,\\C^*_{\mathbf{4c}}&{}\approx{}&-1.076923076923&{}:{}&2.076923076923&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)