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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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&, \\ P_{\mathbf{5}}&{}\approx{}&0.987863392605&{}:{}&0.000376328911&{}:{}&0.011760278483&, \\ P^-_{\mathbf{5}}&{}\approx{}&1.062577447336&{}:{}&-0.031856670797&{}:{}&-0.030720776539&, \\ P^+_{\mathbf{5}}&{}\approx{}&0.925349922240&{}:{}&0.027345775013&{}:{}&0.047304302748&, \\ Q_{\mathbf{5}}&{}\approx{}&0.816326530612&{}:{}&-0.020408163265&{}:{}&0.204081632653&, \\ I^\prime_{\mathbf{5}}&{}\approx{}&0.848484848485&{}:{}&0.020202020202&{}:{}&0.131313131313&, \end{alignedat} \]
\(I\) Incenter
\(P_{\mathbf{5}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{5}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{5}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{5}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{5}}\) Radical center of the Malfatti circles
5 (202)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5}}&{}\approx{}&0.933333333333&{}:{}&0.028344671202&{}:{}&0.038321995465&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.648148148148&{}:{}&-0.952160493827&{}:{}&1.304012345679&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.933333333333&{}:{}&1.388888888889&{}:{}&-1.322222222222&, \\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.765027322404&{}:{}&0.007285974499&{}:{}&0.227686703097&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.933333333333&{}:{}&0.055555555556&{}:{}&0.011111111111&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.925191646841&{}:{}&0.000352453961&{}:{}&0.074455899198&, \\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&-0.530303030303&{}:{}&0.779040404040&{}:{}&0.751262626263&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.998544395924&{}:{}&0.030325084910&{}:{}&-0.028869480835&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.989042675894&{}:{}&-0.029652056901&{}:{}&0.040609381007&, \\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.111111111111&{}:{}&1.111111111111&,\\B^*_{\mathbf{5}}&{}\approx{}&0.800000000000&{}:{}&0.000000000000&{}:{}&0.200000000000&,\\C^*_{\mathbf{5}}&{}\approx{}&1.025641025641&{}:{}&-0.025641025641&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5}}}{B^\prime_{\mathbf{5}}}{C^\prime_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5}}}{B^{\prime\prime}_{\mathbf{5}}}{C^{\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5}}}{B^{\prime\prime\prime}_{\mathbf{5}}}{C^{\prime\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^*_{\mathbf{5}}}{B^*_{\mathbf{5}}}{C^*_{\mathbf{5}}}\)
5 (202)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5}}}}&{}\approx{}&0.085714285714&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5}}}}&{}\approx{}&2.916666666667&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5}}}}&{}\approx{}&4.200000000000&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.933333333333&{}:{}&0.028344671202&{}:{}&0.038321995465&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.648148148148&{}:{}&-0.952160493827&{}:{}&1.304012345679&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.933333333333&{}:{}&1.388888888889&{}:{}&-1.322222222222&. \end{alignedat} \]
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First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5}}&{}\approx{}&0.987863392605&{}:{}&0.000376328911&{}:{}&0.011760278483&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.765027322404&{}:{}&0.007285974499&{}:{}&0.227686703097&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.933333333333&{}:{}&0.055555555556&{}:{}&0.011111111111&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.925191646841&{}:{}&0.000352453961&{}:{}&0.074455899198&. \end{alignedat} \]
5 (202)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5}}&{}\approx{}&1.062577447336&{}:{}&-0.031856670797&{}:{}&-0.030720776539&,\\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&-0.530303030303&{}:{}&0.779040404040&{}:{}&0.751262626263&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.998544395924&{}:{}&0.030325084910&{}:{}&-0.028869480835&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.989042675894&{}:{}&-0.029652056901&{}:{}&0.040609381007&. \end{alignedat} \]
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Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5}}&{}\approx{}&0.925349922240&{}:{}&0.027345775013&{}:{}&0.047304302748&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.933333333333&{}:{}&0.028344671202&{}:{}&0.038321995465&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.648148148148&{}:{}&-0.952160493827&{}:{}&1.304012345679&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.933333333333&{}:{}&1.388888888889&{}:{}&-1.322222222222&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.765027322404&{}:{}&0.007285974499&{}:{}&0.227686703097&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.933333333333&{}:{}&0.055555555556&{}:{}&0.011111111111&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.925191646841&{}:{}&0.000352453961&{}:{}&0.074455899198&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5}}&{}\approx{}&0.816326530612&{}:{}&-0.020408163265&{}:{}&0.204081632653&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.111111111111&{}:{}&1.111111111111&,\\B^*_{\mathbf{5}}&{}\approx{}&0.800000000000&{}:{}&0.000000000000&{}:{}&0.200000000000&,\\C^*_{\mathbf{5}}&{}\approx{}&1.025641025641&{}:{}&-0.025641025641&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5}}&{}\approx{}&0.848484848485&{}:{}&0.020202020202&{}:{}&0.131313131313&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.765027322404&{}:{}&0.007285974499&{}:{}&0.227686703097&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.933333333333&{}:{}&0.055555555556&{}:{}&0.011111111111&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.925191646841&{}:{}&0.000352453961&{}:{}&0.074455899198&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.111111111111&{}:{}&1.111111111111&,\\B^*_{\mathbf{5}}&{}\approx{}&0.800000000000&{}:{}&0.000000000000&{}:{}&0.200000000000&,\\C^*_{\mathbf{5}}&{}\approx{}&1.025641025641&{}:{}&-0.025641025641&{}:{}&0.000000000000&. \end{alignedat} \]
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