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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1a(013)

Malfatti circles

1a (013)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.400000000000&{}:{}&0.595238095238&{}:{}&0.804761904762&, \\ P_{\mathbf{1a}}&{}\approx{}&-0.060074165637&{}:{}&0.915415857319&{}:{}&0.144658308317&, \\ P^-_{\mathbf{1a}}&{}\approx{}&-0.007703281027&{}:{}&0.964744242918&{}:{}&0.042959038109&, \\ P^+_{\mathbf{1a}}&{}\approx{}&-0.100109051254&{}:{}&0.877706807914&{}:{}&0.222402243340&, \\ Q_{\mathbf{1a}}&{}\approx{}&0.096774193548&{}:{}&1.161290322581&{}:{}&-0.258064516129&, \\ I^\prime_{\mathbf{1a}}&{}\approx{}&-0.167701863354&{}:{}&0.798580301686&{}:{}&0.369121561668&, \end{alignedat} \]
\(I_{\mathbf{a}}\) Incenter
\(P_{\mathbf{1a}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{1a}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{1a}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{1a}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{1a}}\) Radical center of the Malfatti circles
1a (013)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1a}}&{}\approx{}&-0.037037037037&{}:{}&0.440917107584&{}:{}&0.596119929453&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.060000000000&{}:{}&0.939285714286&{}:{}&0.120714285714&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-0.843750000000&{}:{}&1.255580357143&{}:{}&0.588169642857&, \\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.112033195021&{}:{}&0.960284528749&{}:{}&0.151748666271&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.246700507614&{}:{}&0.652646845540&{}:{}&0.594053662074&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.056133056133&{}:{}&0.855360855361&{}:{}&0.200772200772&, \\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.065138721351&{}:{}&1.019731173531&{}:{}&0.045407547820&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.060674157303&{}:{}&0.722311396469&{}:{}&0.338362760835&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.007125890736&{}:{}&0.892432982694&{}:{}&0.114692908042&, \\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.285714285714&{}:{}&-0.285714285714&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.600000000000&{}:{}&0.000000000000&{}:{}&1.600000000000&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.076923076923&{}:{}&0.923076923077&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1a}}}{B^\prime_{\mathbf{1a}}}{C^\prime_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^*_{\mathbf{1a}}}{B^*_{\mathbf{1a}}}{C^*_{\mathbf{1a}}}\)
1a (013)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1a}}}}&{}\approx{}&0.740740740741&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1a}}}}&{}\approx{}&0.150000000000&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1a}}}}&{}\approx{}&2.109375000000&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.400000000000&{}:{}&0.595238095238&{}:{}&0.804761904762&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&-0.037037037037&{}:{}&0.440917107584&{}:{}&0.596119929453&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.060000000000&{}:{}&0.939285714286&{}:{}&0.120714285714&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-0.843750000000&{}:{}&1.255580357143&{}:{}&0.588169642857&. \end{alignedat} \]
1a (013)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1a}}&{}\approx{}&-0.060074165637&{}:{}&0.915415857319&{}:{}&0.144658308317&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.112033195021&{}:{}&0.960284528749&{}:{}&0.151748666271&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.246700507614&{}:{}&0.652646845540&{}:{}&0.594053662074&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.056133056133&{}:{}&0.855360855361&{}:{}&0.200772200772&. \end{alignedat} \]
1a (013)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1a}}&{}\approx{}&-0.007703281027&{}:{}&0.964744242918&{}:{}&0.042959038109&,\\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.065138721351&{}:{}&1.019731173531&{}:{}&0.045407547820&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.060674157303&{}:{}&0.722311396469&{}:{}&0.338362760835&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.007125890736&{}:{}&0.892432982694&{}:{}&0.114692908042&. \end{alignedat} \]
1a (013)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1a}}&{}\approx{}&-0.100109051254&{}:{}&0.877706807914&{}:{}&0.222402243340&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&-0.037037037037&{}:{}&0.440917107584&{}:{}&0.596119929453&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.060000000000&{}:{}&0.939285714286&{}:{}&0.120714285714&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-0.843750000000&{}:{}&1.255580357143&{}:{}&0.588169642857&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.112033195021&{}:{}&0.960284528749&{}:{}&0.151748666271&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.246700507614&{}:{}&0.652646845540&{}:{}&0.594053662074&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.056133056133&{}:{}&0.855360855361&{}:{}&0.200772200772&, \end{alignedat} \]
1a (013)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1a}}&{}\approx{}&0.096774193548&{}:{}&1.161290322581&{}:{}&-0.258064516129&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.285714285714&{}:{}&-0.285714285714&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.600000000000&{}:{}&0.000000000000&{}:{}&1.600000000000&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.076923076923&{}:{}&0.923076923077&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1a}}&{}\approx{}&-0.167701863354&{}:{}&0.798580301686&{}:{}&0.369121561668&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.112033195021&{}:{}&0.960284528749&{}:{}&0.151748666271&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.246700507614&{}:{}&0.652646845540&{}:{}&0.594053662074&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.056133056133&{}:{}&0.855360855361&{}:{}&0.200772200772&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.285714285714&{}:{}&-0.285714285714&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.600000000000&{}:{}&0.000000000000&{}:{}&1.600000000000&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.076923076923&{}:{}&0.923076923077&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)