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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3a (033)

Malfatti circles

3a (033)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.400000000000&{}:{}&0.595238095238&{}:{}&0.804761904762&, \\ P_{\mathbf{3a}}&{}\approx{}&-0.004778156997&{}:{}&0.072810011377&{}:{}&0.931968145620&, \\ P^-_{\mathbf{3a}}&{}\approx{}&0.036981132075&{}:{}&0.017610062893&{}:{}&0.945408805031&, \\ P^+_{\mathbf{3a}}&{}\approx{}&-0.039252336449&{}:{}&0.118380062305&{}:{}&0.920872274143&, \\ Q_{\mathbf{3a}}&{}\approx{}&0.027027027027&{}:{}&0.324324324324&{}:{}&0.648648648649&, \\ I^\prime_{\mathbf{3a}}&{}\approx{}&-0.042424242424&{}:{}&0.202020202020&{}:{}&0.840404040404&, \end{alignedat} \]
\(I_{\mathbf{a}}\) Incenter
\(P_{\mathbf{3a}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{3a}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{3a}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{3a}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{3a}}\) Radical center of the Malfatti circles
3a (033)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3a}}&{}\approx{}&0.200000000000&{}:{}&0.340136054422&{}:{}&0.459863945578&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.700000000000&{}:{}&0.291666666667&{}:{}&1.408333333333&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.043750000000&{}:{}&0.065104166667&{}:{}&0.978645833333&, \\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.082352941176&{}:{}&0.078431372549&{}:{}&1.003921568627&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.004590163934&{}:{}&0.109289617486&{}:{}&0.895300546448&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.021538461538&{}:{}&0.328205128205&{}:{}&0.693333333333&, \\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.045901639344&{}:{}&0.019125683060&{}:{}&1.026775956284&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.035379061372&{}:{}&0.060168471721&{}:{}&0.904452466907&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.264864864865&{}:{}&0.126126126126&{}:{}&0.609009009009&, \\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.333333333333&{}:{}&0.666666666667&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.040000000000&{}:{}&0.000000000000&{}:{}&0.960000000000&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.076923076923&{}:{}&0.923076923077&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3a}}}{B^\prime_{\mathbf{3a}}}{C^\prime_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^*_{\mathbf{3a}}}{B^*_{\mathbf{3a}}}{C^*_{\mathbf{3a}}}\)
3a (033)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3a}}}}&{}\approx{}&0.571428571429&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3a}}}}&{}\approx{}&1.750000000000&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3a}}}}&{}\approx{}&0.109375000000&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.400000000000&{}:{}&0.595238095238&{}:{}&0.804761904762&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.200000000000&{}:{}&0.340136054422&{}:{}&0.459863945578&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.700000000000&{}:{}&0.291666666667&{}:{}&1.408333333333&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.043750000000&{}:{}&0.065104166667&{}:{}&0.978645833333&. \end{alignedat} \]
3a (033)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3a}}&{}\approx{}&-0.004778156997&{}:{}&0.072810011377&{}:{}&0.931968145620&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.082352941176&{}:{}&0.078431372549&{}:{}&1.003921568627&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.004590163934&{}:{}&0.109289617486&{}:{}&0.895300546448&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.021538461538&{}:{}&0.328205128205&{}:{}&0.693333333333&. \end{alignedat} \]
3a (033)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3a}}&{}\approx{}&0.036981132075&{}:{}&0.017610062893&{}:{}&0.945408805031&,\\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.045901639344&{}:{}&0.019125683060&{}:{}&1.026775956284&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.035379061372&{}:{}&0.060168471721&{}:{}&0.904452466907&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.264864864865&{}:{}&0.126126126126&{}:{}&0.609009009009&. \end{alignedat} \]
3a (033)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3a}}&{}\approx{}&-0.039252336449&{}:{}&0.118380062305&{}:{}&0.920872274143&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.200000000000&{}:{}&0.340136054422&{}:{}&0.459863945578&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.700000000000&{}:{}&0.291666666667&{}:{}&1.408333333333&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.043750000000&{}:{}&0.065104166667&{}:{}&0.978645833333&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.082352941176&{}:{}&0.078431372549&{}:{}&1.003921568627&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.004590163934&{}:{}&0.109289617486&{}:{}&0.895300546448&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.021538461538&{}:{}&0.328205128205&{}:{}&0.693333333333&, \end{alignedat} \]
3a (033)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3a}}&{}\approx{}&0.027027027027&{}:{}&0.324324324324&{}:{}&0.648648648649&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.333333333333&{}:{}&0.666666666667&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.040000000000&{}:{}&0.000000000000&{}:{}&0.960000000000&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.076923076923&{}:{}&0.923076923077&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3a}}&{}\approx{}&-0.042424242424&{}:{}&0.202020202020&{}:{}&0.840404040404&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.082352941176&{}:{}&0.078431372549&{}:{}&1.003921568627&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.004590163934&{}:{}&0.109289617486&{}:{}&0.895300546448&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.021538461538&{}:{}&0.328205128205&{}:{}&0.693333333333&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.333333333333&{}:{}&0.666666666667&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.040000000000&{}:{}&0.000000000000&{}:{}&0.960000000000&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.076923076923&{}:{}&0.923076923077&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)