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,
\[ \begin{alignedat}{4} I&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&, \\ P&{}\approx{}&0.178316690442&{}:{}&0.278241967258&{}:{}&0.543441342300&, \\ P^-&{}\approx{}&0.163147792706&{}:{}&0.260122475094&{}:{}&0.576729732200&, \\ P^+&{}\approx{}&0.187287173666&{}:{}&0.288957353657&{}:{}&0.523755472677&, \\ Q&{}\approx{}&0.151515151515&{}:{}&0.242424242424&{}:{}&0.606060606061&, \\ I^\prime&{}\approx{}&0.200000000000&{}:{}&0.304761904762&{}:{}&0.495238095238&, \end{alignedat} \]
\(I\) Incenter
\(P\) First Ajima-Malfatti point
\(P^-\) First Malfatti-Rabinowitz point
\(P^+\) Gergonne point of the Malfatti triangle
\(Q\) Second Malfatti-Rabinowitz point
\(I^\prime\) Radical center of the Malfatti circles
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.377777777778&{}:{}&0.264550264550&{}:{}&0.357671957672&,\\B^\prime&{}\approx{}&0.156250000000&{}:{}&0.529389880952&{}:{}&0.314360119048&,\\C^\prime&{}\approx{}&0.100000000000&{}:{}&0.148809523810&{}:{}&0.751190476190&, \\ A^{\prime\prime}&{}\approx{}&0.121951219512&{}:{}&0.297328687573&{}:{}&0.580720092915&,\\B^{\prime\prime}&{}\approx{}&0.200000000000&{}:{}&0.190476190476&{}:{}&0.609523809524&,\\C^{\prime\prime}&{}\approx{}&0.259875259875&{}:{}&0.405504405504&{}:{}&0.334620334620&, \\ A^{\prime\prime\prime}&{}\approx{}&0.084033613445&{}:{}&0.284713885554&{}:{}&0.631252501000&,\\B^{\prime\prime\prime}&{}\approx{}&0.191011235955&{}:{}&0.133761369716&{}:{}&0.675227394329&,\\C^{\prime\prime\prime}&{}\approx{}&0.282392026578&{}:{}&0.450245214365&{}:{}&0.267362759057&, \\ A^*&{}\approx{}&0.000000000000&{}:{}&0.285714285714&{}:{}&0.714285714286&,\\B^*&{}\approx{}&0.200000000000&{}:{}&0.000000000000&{}:{}&0.800000000000&,\\C^*&{}\approx{}&0.384615384615&{}:{}&0.615384615385&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime}{B^\prime}{C^\prime}\)
\(\triangle{A^{\prime\prime}}{B^{\prime\prime}}{C^{\prime\prime}}\)
\(\triangle{A^{\prime\prime\prime}}{B^{\prime\prime\prime}}{C^{\prime\prime\prime}}\)
\(\triangle{A^*}{B^*}{C^*}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime}}&{}\approx{}&0.800000000000&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime}}&{}\approx{}&0.703125000000&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime}}&{}\approx{}&0.450000000000&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&,\\ A^\prime&{}\approx{}&0.377777777778&{}:{}&0.264550264550&{}:{}&0.357671957672&,\\B^\prime&{}\approx{}&0.156250000000&{}:{}&0.529389880952&{}:{}&0.314360119048&,\\C^\prime&{}\approx{}&0.100000000000&{}:{}&0.148809523810&{}:{}&0.751190476190&. \end{alignedat} \]
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First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P&{}\approx{}&0.178316690442&{}:{}&0.278241967258&{}:{}&0.543441342300&,\\ A^{\prime\prime}&{}\approx{}&0.121951219512&{}:{}&0.297328687573&{}:{}&0.580720092915&,\\B^{\prime\prime}&{}\approx{}&0.200000000000&{}:{}&0.190476190476&{}:{}&0.609523809524&,\\C^{\prime\prime}&{}\approx{}&0.259875259875&{}:{}&0.405504405504&{}:{}&0.334620334620&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-&{}\approx{}&0.163147792706&{}:{}&0.260122475094&{}:{}&0.576729732200&,\\ A^{\prime\prime\prime}&{}\approx{}&0.084033613445&{}:{}&0.284713885554&{}:{}&0.631252501000&,\\B^{\prime\prime\prime}&{}\approx{}&0.191011235955&{}:{}&0.133761369716&{}:{}&0.675227394329&,\\C^{\prime\prime\prime}&{}\approx{}&0.282392026578&{}:{}&0.450245214365&{}:{}&0.267362759057&. \end{alignedat} \]
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Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+&{}\approx{}&0.187287173666&{}:{}&0.288957353657&{}:{}&0.523755472677&,\\ A^\prime&{}\approx{}&0.377777777778&{}:{}&0.264550264550&{}:{}&0.357671957672&,\\B^\prime&{}\approx{}&0.156250000000&{}:{}&0.529389880952&{}:{}&0.314360119048&,\\C^\prime&{}\approx{}&0.100000000000&{}:{}&0.148809523810&{}:{}&0.751190476190&,\\ A^{\prime\prime}&{}\approx{}&0.121951219512&{}:{}&0.297328687573&{}:{}&0.580720092915&,\\B^{\prime\prime}&{}\approx{}&0.200000000000&{}:{}&0.190476190476&{}:{}&0.609523809524&,\\C^{\prime\prime}&{}\approx{}&0.259875259875&{}:{}&0.405504405504&{}:{}&0.334620334620&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q&{}\approx{}&0.151515151515&{}:{}&0.242424242424&{}:{}&0.606060606061&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.285714285714&{}:{}&0.714285714286&,\\B^*&{}\approx{}&0.200000000000&{}:{}&0.000000000000&{}:{}&0.800000000000&,\\C^*&{}\approx{}&0.384615384615&{}:{}&0.615384615385&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.200000000000&{}:{}&0.304761904762&{}:{}&0.495238095238&,\\ A^{\prime\prime}&{}\approx{}&0.121951219512&{}:{}&0.297328687573&{}:{}&0.580720092915&,\\B^{\prime\prime}&{}\approx{}&0.200000000000&{}:{}&0.190476190476&{}:{}&0.609523809524&,\\C^{\prime\prime}&{}\approx{}&0.259875259875&{}:{}&0.405504405504&{}:{}&0.334620334620&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.285714285714&{}:{}&0.714285714286&,\\B^*&{}\approx{}&0.200000000000&{}:{}&0.000000000000&{}:{}&0.800000000000&,\\C^*&{}\approx{}&0.384615384615&{}:{}&0.615384615385&{}:{}&0.000000000000&. \end{alignedat} \]
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