Derousseau's Generalization of the Malfatti circles

Martin's solution

Problem 4331 (proposed by A. Martin) I. Solution by the Proposer, Mathematical Questions with their Solutions, from the “Educational Times”.

\(a:b:c=231:250:289\).

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5a (213)

Malfatti circles

5a (213)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&, \\ P_{\mathbf{5a}}&{}\approx{}&1.079560324522&{}:{}&-0.004785583430&{}:{}&-0.074774741092&, \\ P^-_{\mathbf{5a}}&{}\approx{}&0.890636000939&{}:{}&0.079525262346&{}:{}&0.029838736715&, \\ P^+_{\mathbf{5a}}&{}\approx{}&1.317657497782&{}:{}&-0.111040689568&{}:{}&-0.206616808214&, \\ Q_{\mathbf{5a}}&{}\approx{}&1.451612903226&{}:{}&0.193548387097&{}:{}&-0.645161290323&, \\ I^\prime_{\mathbf{5a}}&{}\approx{}&1.571428571429&{}:{}&-0.108843537415&{}:{}&-0.462585034014&, \end{alignedat} \]
\(I_{\mathbf{a}}\) Incenter
\(P_{\mathbf{5a}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{5a}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{5a}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{5a}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{5a}}\) Radical center of the Malfatti circles
5a (213)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5a}}&{}\approx{}&1.254545454545&{}:{}&-0.118063754427&{}:{}&-0.136481700118&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&1.289062500000&{}:{}&1.323660714286&{}:{}&-1.612723214286&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&3.300000000000&{}:{}&-3.571428571429&{}:{}&1.271428571429&, \\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.853932584270&{}:{}&-0.051364365971&{}:{}&-0.802568218299&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.320000000000&{}:{}&-0.228571428571&{}:{}&-0.091428571429&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.257238646754&{}:{}&-0.005573213741&{}:{}&-0.251665433013&, \\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.414572864322&{}:{}&0.425699928212&{}:{}&0.159727207466&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.064516129032&{}:{}&-0.100180324584&{}:{}&0.035664195552&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.019887127116&{}:{}&0.091066149653&{}:{}&-0.110953276769&, \\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.428571428571&{}:{}&1.428571428571&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.800000000000&{}:{}&0.000000000000&{}:{}&-0.800000000000&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.882352941176&{}:{}&0.117647058824&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5a}}}{B^\prime_{\mathbf{5a}}}{C^\prime_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^*_{\mathbf{5a}}}{B^*_{\mathbf{5a}}}{C^*_{\mathbf{5a}}}\)
5a (213)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5a}}}}&{}\approx{}&-0.145454545455&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5a}}}}&{}\approx{}&-1.718750000000&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5a}}}}&{}\approx{}&-4.400000000000&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.254545454545&{}:{}&-0.118063754427&{}:{}&-0.136481700118&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&1.289062500000&{}:{}&1.323660714286&{}:{}&-1.612723214286&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&3.300000000000&{}:{}&-3.571428571429&{}:{}&1.271428571429&. \end{alignedat} \]
5a (213)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5a}}&{}\approx{}&1.079560324522&{}:{}&-0.004785583430&{}:{}&-0.074774741092&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.853932584270&{}:{}&-0.051364365971&{}:{}&-0.802568218299&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.320000000000&{}:{}&-0.228571428571&{}:{}&-0.091428571429&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.257238646754&{}:{}&-0.005573213741&{}:{}&-0.251665433013&. \end{alignedat} \]
5a (213)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5a}}&{}\approx{}&0.890636000939&{}:{}&0.079525262346&{}:{}&0.029838736715&,\\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.414572864322&{}:{}&0.425699928212&{}:{}&0.159727207466&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.064516129032&{}:{}&-0.100180324584&{}:{}&0.035664195552&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.019887127116&{}:{}&0.091066149653&{}:{}&-0.110953276769&. \end{alignedat} \]
5a (213)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5a}}&{}\approx{}&1.317657497782&{}:{}&-0.111040689568&{}:{}&-0.206616808214&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.254545454545&{}:{}&-0.118063754427&{}:{}&-0.136481700118&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&1.289062500000&{}:{}&1.323660714286&{}:{}&-1.612723214286&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&3.300000000000&{}:{}&-3.571428571429&{}:{}&1.271428571429&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.853932584270&{}:{}&-0.051364365971&{}:{}&-0.802568218299&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.320000000000&{}:{}&-0.228571428571&{}:{}&-0.091428571429&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.257238646754&{}:{}&-0.005573213741&{}:{}&-0.251665433013&, \end{alignedat} \]
5a (213)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5a}}&{}\approx{}&1.451612903226&{}:{}&0.193548387097&{}:{}&-0.645161290323&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.428571428571&{}:{}&1.428571428571&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.800000000000&{}:{}&0.000000000000&{}:{}&-0.800000000000&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.882352941176&{}:{}&0.117647058824&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5a}}&{}\approx{}&1.571428571429&{}:{}&-0.108843537415&{}:{}&-0.462585034014&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.853932584270&{}:{}&-0.051364365971&{}:{}&-0.802568218299&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.320000000000&{}:{}&-0.228571428571&{}:{}&-0.091428571429&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.257238646754&{}:{}&-0.005573213741&{}:{}&-0.251665433013&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.428571428571&{}:{}&1.428571428571&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.800000000000&{}:{}&0.000000000000&{}:{}&-0.800000000000&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.882352941176&{}:{}&0.117647058824&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)