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,
\[ \begin{alignedat}{4} I&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&, \\ P_{\mathbf{3}}&{}\approx{}&0.000975609756&{}:{}&0.951451800232&{}:{}&0.047572590012&, \\ P^-_{\mathbf{3}}&{}\approx{}&-0.035187287174&{}:{}&1.052916058591&{}:{}&-0.017728771418&, \\ P^+_{\mathbf{3}}&{}\approx{}&0.028229255774&{}:{}&0.874984724429&{}:{}&0.096786019797&, \\ Q_{\mathbf{3}}&{}\approx{}&-0.018181818182&{}:{}&0.727272727273&{}:{}&0.290909090909&, \\ I^\prime_{\mathbf{3}}&{}\approx{}&0.020408163265&{}:{}&0.777453838678&{}:{}&0.202137998056&, \end{alignedat} \]
\(I\) Incenter
\(P_{\mathbf{3}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{3}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{3}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{3}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{3}}\) Radical center of the Malfatti circles
3 (022)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3}}&{}\approx{}&-0.382716049383&{}:{}&0.587889476778&{}:{}&0.794826572604&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.031250000000&{}:{}&0.905877976190&{}:{}&0.062872023810&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.500000000000&{}:{}&0.744047619048&{}:{}&-0.244047619048&, \\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.058823529412&{}:{}&0.896358543417&{}:{}&0.044817927171&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006896551724&{}:{}&0.656814449918&{}:{}&0.336288998358&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000904977376&{}:{}&0.882568411980&{}:{}&0.116526610644&, \\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.033898305085&{}:{}&0.982647296207&{}:{}&-0.016545601291&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-31.000000000006&{}:{}&47.619047619057&{}:{}&-15.619047619051&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.032258064516&{}:{}&0.965264357564&{}:{}&0.066993706952&, \\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.714285714286&{}:{}&0.285714285714&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.066666666667&{}:{}&0.000000000000&{}:{}&1.066666666667&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.025641025641&{}:{}&1.025641025641&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3}}}{B^\prime_{\mathbf{3}}}{C^\prime_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3}}}{B^{\prime\prime}_{\mathbf{3}}}{C^{\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3}}}{B^{\prime\prime\prime}_{\mathbf{3}}}{C^{\prime\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^*_{\mathbf{3}}}{B^*_{\mathbf{3}}}{C^*_{\mathbf{3}}}\)
3 (022)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3}}}}&{}\approx{}&1.777777777778&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3}}}}&{}\approx{}&0.140625000000&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3}}}}&{}\approx{}&2.250000000000&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.382716049383&{}:{}&0.587889476778&{}:{}&0.794826572604&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.031250000000&{}:{}&0.905877976190&{}:{}&0.062872023810&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.500000000000&{}:{}&0.744047619048&{}:{}&-0.244047619048&. \end{alignedat} \]
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First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3}}&{}\approx{}&0.000975609756&{}:{}&0.951451800232&{}:{}&0.047572590012&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.058823529412&{}:{}&0.896358543417&{}:{}&0.044817927171&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006896551724&{}:{}&0.656814449918&{}:{}&0.336288998358&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000904977376&{}:{}&0.882568411980&{}:{}&0.116526610644&. \end{alignedat} \]
3 (022)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3}}&{}\approx{}&-0.035187287174&{}:{}&1.052916058591&{}:{}&-0.017728771418&,\\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.033898305085&{}:{}&0.982647296207&{}:{}&-0.016545601291&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-31.000000000006&{}:{}&47.619047619057&{}:{}&-15.619047619051&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.032258064516&{}:{}&0.965264357564&{}:{}&0.066993706952&. \end{alignedat} \]
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Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3}}&{}\approx{}&0.028229255774&{}:{}&0.874984724429&{}:{}&0.096786019797&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.382716049383&{}:{}&0.587889476778&{}:{}&0.794826572604&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.031250000000&{}:{}&0.905877976190&{}:{}&0.062872023810&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.500000000000&{}:{}&0.744047619048&{}:{}&-0.244047619048&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.058823529412&{}:{}&0.896358543417&{}:{}&0.044817927171&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006896551724&{}:{}&0.656814449918&{}:{}&0.336288998358&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000904977376&{}:{}&0.882568411980&{}:{}&0.116526610644&, \end{alignedat} \]
3 (022)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3}}&{}\approx{}&-0.018181818182&{}:{}&0.727272727273&{}:{}&0.290909090909&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.714285714286&{}:{}&0.285714285714&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.066666666667&{}:{}&0.000000000000&{}:{}&1.066666666667&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.025641025641&{}:{}&1.025641025641&{}:{}&0.000000000000&. \end{alignedat} \]
3 (022)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3}}&{}\approx{}&0.020408163265&{}:{}&0.777453838678&{}:{}&0.202137998056&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.058823529412&{}:{}&0.896358543417&{}:{}&0.044817927171&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006896551724&{}:{}&0.656814449918&{}:{}&0.336288998358&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000904977376&{}:{}&0.882568411980&{}:{}&0.116526610644&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.714285714286&{}:{}&0.285714285714&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.066666666667&{}:{}&0.000000000000&{}:{}&1.066666666667&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.025641025641&{}:{}&1.025641025641&{}:{}&0.000000000000&. \end{alignedat} \]
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