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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2a(031)

Malfatti circles

2a (031)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.400000000000&{}:{}&0.595238095238&{}:{}&0.804761904762&, \\ P_{\mathbf{2a}}&{}\approx{}&-0.005074658035&{}:{}&0.015274690852&{}:{}&0.989799967183&, \\ P^-_{\mathbf{2a}}&{}\approx{}&0.013581191908&{}:{}&-0.012122158869&{}:{}&0.998540966961&, \\ P^+_{\mathbf{2a}}&{}\approx{}&-0.022120091917&{}:{}&0.040306581220&{}:{}&0.981813510698&, \\ Q_{\mathbf{2a}}&{}\approx{}&0.050847457627&{}:{}&-0.271186440678&{}:{}&1.220338983051&, \\ I^\prime_{\mathbf{2a}}&{}\approx{}&-0.047787610619&{}:{}&0.101137800253&{}:{}&0.946649810367&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{2a}}\)
\(P^-_{\mathbf{2a}}\)
\(P^+_{\mathbf{2a}}\)
\(Q_{\mathbf{2a}}\)
\(I^\prime_{\mathbf{2a}}\)
2a (031)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2a}}&{}\approx{}&0.170370370370&{}:{}&0.352733686067&{}:{}&0.476895943563&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-2.700000000000&{}:{}&-1.732142857143&{}:{}&5.432142857143&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.018750000000&{}:{}&0.027901785714&{}:{}&0.990848214286&, \\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.037241379310&{}:{}&0.015763546798&{}:{}&1.021477832512&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.004886877828&{}:{}&0.051712992889&{}:{}&0.953173884939&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.061286254729&{}:{}&0.184471266438&{}:{}&0.876814988290&, \\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.019529837251&{}:{}&-0.012529062258&{}:{}&1.032058899509&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.013055818354&{}:{}&0.027030679822&{}:{}&0.959913501825&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.344044321330&{}:{}&-0.307083498219&{}:{}&0.963039176890&, \\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.285714285714&{}:{}&1.285714285714&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.040000000000&{}:{}&0.000000000000&{}:{}&0.960000000000&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.230769230769&{}:{}&1.230769230769&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2a}}}{B^\prime_{\mathbf{2a}}}{C^\prime_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^*_{\mathbf{2a}}}{B^*_{\mathbf{2a}}}{C^*_{\mathbf{2a}}}\)
2a (031)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2a}}}}&{}\approx{}&0.592592592593&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2a}}}}&{}\approx{}&6.750000000000&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2a}}}}&{}\approx{}&0.046875000000&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.400000000000&{}:{}&0.595238095238&{}:{}&0.804761904762&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.170370370370&{}:{}&0.352733686067&{}:{}&0.476895943563&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-2.700000000000&{}:{}&-1.732142857143&{}:{}&5.432142857143&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.018750000000&{}:{}&0.027901785714&{}:{}&0.990848214286&. \end{alignedat} \]
2a (031)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2a}}&{}\approx{}&-0.005074658035&{}:{}&0.015274690852&{}:{}&0.989799967183&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.037241379310&{}:{}&0.015763546798&{}:{}&1.021477832512&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.004886877828&{}:{}&0.051712992889&{}:{}&0.953173884939&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.061286254729&{}:{}&0.184471266438&{}:{}&0.876814988290&. \end{alignedat} \]
2a (031)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2a}}&{}\approx{}&0.013581191908&{}:{}&-0.012122158869&{}:{}&0.998540966961&,\\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.019529837251&{}:{}&-0.012529062258&{}:{}&1.032058899509&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.013055818354&{}:{}&0.027030679822&{}:{}&0.959913501825&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.344044321330&{}:{}&-0.307083498219&{}:{}&0.963039176890&. \end{alignedat} \]
2a (031)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2a}}&{}\approx{}&-0.022120091917&{}:{}&0.040306581220&{}:{}&0.981813510698&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.170370370370&{}:{}&0.352733686067&{}:{}&0.476895943563&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-2.700000000000&{}:{}&-1.732142857143&{}:{}&5.432142857143&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.018750000000&{}:{}&0.027901785714&{}:{}&0.990848214286&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.037241379310&{}:{}&0.015763546798&{}:{}&1.021477832512&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.004886877828&{}:{}&0.051712992889&{}:{}&0.953173884939&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.061286254729&{}:{}&0.184471266438&{}:{}&0.876814988290&, \end{alignedat} \]
2a (031)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2a}}&{}\approx{}&0.050847457627&{}:{}&-0.271186440678&{}:{}&1.220338983051&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.285714285714&{}:{}&1.285714285714&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.040000000000&{}:{}&0.000000000000&{}:{}&0.960000000000&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.230769230769&{}:{}&1.230769230769&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2a}}&{}\approx{}&-0.047787610619&{}:{}&0.101137800253&{}:{}&0.946649810367&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.037241379310&{}:{}&0.015763546798&{}:{}&1.021477832512&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.004886877828&{}:{}&0.051712992889&{}:{}&0.953173884939&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.061286254729&{}:{}&0.184471266438&{}:{}&0.876814988290&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.285714285714&{}:{}&1.285714285714&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.040000000000&{}:{}&0.000000000000&{}:{}&0.960000000000&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.230769230769&{}:{}&1.230769230769&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)