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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6 (220)

Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&, \\ P_{\mathbf{6}}&{}\approx{}&0.807207207207&{}:{}&0.192192192192&{}:{}&0.000600600601&, \\ P^-_{\mathbf{6}}&{}\approx{}&1.293729372937&{}:{}&0.077007700770&{}:{}&-0.370737073707&, \\ P^+_{\mathbf{6}}&{}\approx{}&0.624535315985&{}:{}&0.235439900867&{}:{}&0.140024783147&, \\ Q_{\mathbf{6}}&{}\approx{}&0.640000000000&{}:{}&0.400000000000&{}:{}&-0.040000000000&, \\ I^\prime_{\mathbf{6}}&{}\approx{}&0.612021857923&{}:{}&0.364298724954&{}:{}&0.023679417122&, \end{alignedat} \]
\(I\) Incenter
\(P_{\mathbf{6}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{6}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{6}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{6}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{6}}\) Radical center of the Malfatti circles
6 (220)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6}}&{}\approx{}&0.666666666667&{}:{}&0.141723356009&{}:{}&0.191609977324&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.291666666667&{}:{}&0.121527777778&{}:{}&0.586805555556&,\\C^\prime_{\mathbf{6}}&{}\approx{}&2.074074074074&{}:{}&3.086419753086&{}:{}&-4.160493827160&, \\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.511415525114&{}:{}&0.487062404871&{}:{}&0.001522070015&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.728455284553&{}:{}&0.271002710027&{}:{}&0.000542005420&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.574358974359&{}:{}&0.136752136752&{}:{}&0.288888888889&, \\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&-1.696969696970&{}:{}&-0.707070707071&{}:{}&3.404040404040&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.079889807163&{}:{}&0.229568411387&{}:{}&-0.309458218549&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.742424242424&{}:{}&0.044191919192&{}:{}&0.213383838384&, \\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.111111111111&{}:{}&-0.111111111111&,\\B^*_{\mathbf{6}}&{}\approx{}&1.066666666667&{}:{}&0.000000000000&{}:{}&-0.066666666667&,\\C^*_{\mathbf{6}}&{}\approx{}&0.615384615385&{}:{}&0.384615384615&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6}}}{B^\prime_{\mathbf{6}}}{C^\prime_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6}}}{B^{\prime\prime}_{\mathbf{6}}}{C^{\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6}}}{B^{\prime\prime\prime}_{\mathbf{6}}}{C^{\prime\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^*_{\mathbf{6}}}{B^*_{\mathbf{6}}}{C^*_{\mathbf{6}}}\)
6 (220)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6}}}}&{}\approx{}&0.428571428571&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6}}}}&{}\approx{}&1.312500000000&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6}}}}&{}\approx{}&9.333333333333&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.666666666667&{}:{}&0.141723356009&{}:{}&0.191609977324&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.291666666667&{}:{}&0.121527777778&{}:{}&0.586805555556&,\\C^\prime_{\mathbf{6}}&{}\approx{}&2.074074074074&{}:{}&3.086419753086&{}:{}&-4.160493827160&. \end{alignedat} \]
6 (220)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6}}&{}\approx{}&0.807207207207&{}:{}&0.192192192192&{}:{}&0.000600600601&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.511415525114&{}:{}&0.487062404871&{}:{}&0.001522070015&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.728455284553&{}:{}&0.271002710027&{}:{}&0.000542005420&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.574358974359&{}:{}&0.136752136752&{}:{}&0.288888888889&. \end{alignedat} \]
6 (220)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6}}&{}\approx{}&1.293729372937&{}:{}&0.077007700770&{}:{}&-0.370737073707&,\\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&-1.696969696970&{}:{}&-0.707070707071&{}:{}&3.404040404040&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.079889807163&{}:{}&0.229568411387&{}:{}&-0.309458218549&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.742424242424&{}:{}&0.044191919192&{}:{}&0.213383838384&. \end{alignedat} \]
6 (220)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6}}&{}\approx{}&0.624535315985&{}:{}&0.235439900867&{}:{}&0.140024783147&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.666666666667&{}:{}&0.141723356009&{}:{}&0.191609977324&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.291666666667&{}:{}&0.121527777778&{}:{}&0.586805555556&,\\C^\prime_{\mathbf{6}}&{}\approx{}&2.074074074074&{}:{}&3.086419753086&{}:{}&-4.160493827160&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.511415525114&{}:{}&0.487062404871&{}:{}&0.001522070015&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.728455284553&{}:{}&0.271002710027&{}:{}&0.000542005420&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.574358974359&{}:{}&0.136752136752&{}:{}&0.288888888889&, \end{alignedat} \]
6 (220)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6}}&{}\approx{}&0.640000000000&{}:{}&0.400000000000&{}:{}&-0.040000000000&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.111111111111&{}:{}&-0.111111111111&,\\B^*_{\mathbf{6}}&{}\approx{}&1.066666666667&{}:{}&0.000000000000&{}:{}&-0.066666666667&,\\C^*_{\mathbf{6}}&{}\approx{}&0.615384615385&{}:{}&0.384615384615&{}:{}&0.000000000000&. \end{alignedat} \]
6 (220)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6}}&{}\approx{}&0.612021857923&{}:{}&0.364298724954&{}:{}&0.023679417122&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.511415525114&{}:{}&0.487062404871&{}:{}&0.001522070015&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.728455284553&{}:{}&0.271002710027&{}:{}&0.000542005420&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.574358974359&{}:{}&0.136752136752&{}:{}&0.288888888889&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.111111111111&{}:{}&-0.111111111111&,\\B^*_{\mathbf{6}}&{}\approx{}&1.066666666667&{}:{}&0.000000000000&{}:{}&-0.066666666667&,\\C^*_{\mathbf{6}}&{}\approx{}&0.615384615385&{}:{}&0.384615384615&{}:{}&0.000000000000&. \end{alignedat} \]
6 (220)