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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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&, \\ P_{\mathbf{5}}&{}\approx{}&0.781395348837&{}:{}&0.002114164905&{}:{}&0.216490486258&, \\ P^-_{\mathbf{5}}&{}\approx{}&1.013793103448&{}:{}&-0.153605015674&{}:{}&0.139811912226&, \\ P^+_{\mathbf{5}}&{}\approx{}&0.663157894737&{}:{}&0.081339712919&{}:{}&0.255502392344&, \\ Q_{\mathbf{5}}&{}\approx{}&0.620689655172&{}:{}&-0.034482758621&{}:{}&0.413793103448&, \\ I^\prime_{\mathbf{5}}&{}\approx{}&0.608695652174&{}:{}&0.032938076416&{}:{}&0.358366271410&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{5}}\)
\(P^-_{\mathbf{5}}\)
\(P^+_{\mathbf{5}}\)
\(Q_{\mathbf{5}}\)
\(I^\prime_{\mathbf{5}}\)
5 (202)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5}}&{}\approx{}&0.750000000000&{}:{}&0.115955473098&{}:{}&0.134044526902&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.840000000000&{}:{}&-0.890909090909&{}:{}&1.050909090909&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.328125000000&{}:{}&0.355113636364&{}:{}&0.316761363636&, \\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.471910112360&{}:{}&0.005107252298&{}:{}&0.522982635342&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.646153846154&{}:{}&0.174825174825&{}:{}&0.179020979021&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.760180995475&{}:{}&0.002056766763&{}:{}&0.237762237762&, \\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&1.105263157895&{}:{}&-1.172248803828&{}:{}&1.066985645933&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.773684210526&{}:{}&0.119617224880&{}:{}&0.106698564593&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.973509933775&{}:{}&-0.147501505117&{}:{}&0.173991571343&, \\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.090909090909&{}:{}&1.090909090909&,\\B^*_{\mathbf{5}}&{}\approx{}&0.600000000000&{}:{}&0.000000000000&{}:{}&0.400000000000&,\\C^*_{\mathbf{5}}&{}\approx{}&1.058823529412&{}:{}&-0.058823529412&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5}}}{B^\prime_{\mathbf{5}}}{C^\prime_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5}}}{B^{\prime\prime}_{\mathbf{5}}}{C^{\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5}}}{B^{\prime\prime\prime}_{\mathbf{5}}}{C^{\prime\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^*_{\mathbf{5}}}{B^*_{\mathbf{5}}}{C^*_{\mathbf{5}}}\)
5 (202)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5}}}}&{}\approx{}&0.357142857143&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5}}}}&{}\approx{}&2.800000000000&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5}}}}&{}\approx{}&1.093750000000&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.750000000000&{}:{}&0.115955473098&{}:{}&0.134044526902&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.840000000000&{}:{}&-0.890909090909&{}:{}&1.050909090909&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.328125000000&{}:{}&0.355113636364&{}:{}&0.316761363636&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5}}&{}\approx{}&0.781395348837&{}:{}&0.002114164905&{}:{}&0.216490486258&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.471910112360&{}:{}&0.005107252298&{}:{}&0.522982635342&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.646153846154&{}:{}&0.174825174825&{}:{}&0.179020979021&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.760180995475&{}:{}&0.002056766763&{}:{}&0.237762237762&. \end{alignedat} \]
5 (202)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5}}&{}\approx{}&1.013793103448&{}:{}&-0.153605015674&{}:{}&0.139811912226&,\\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&1.105263157895&{}:{}&-1.172248803828&{}:{}&1.066985645933&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.773684210526&{}:{}&0.119617224880&{}:{}&0.106698564593&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.973509933775&{}:{}&-0.147501505117&{}:{}&0.173991571343&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5}}&{}\approx{}&0.663157894737&{}:{}&0.081339712919&{}:{}&0.255502392344&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.750000000000&{}:{}&0.115955473098&{}:{}&0.134044526902&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.840000000000&{}:{}&-0.890909090909&{}:{}&1.050909090909&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.328125000000&{}:{}&0.355113636364&{}:{}&0.316761363636&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.471910112360&{}:{}&0.005107252298&{}:{}&0.522982635342&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.646153846154&{}:{}&0.174825174825&{}:{}&0.179020979021&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.760180995475&{}:{}&0.002056766763&{}:{}&0.237762237762&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5}}&{}\approx{}&0.620689655172&{}:{}&-0.034482758621&{}:{}&0.413793103448&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.090909090909&{}:{}&1.090909090909&,\\B^*_{\mathbf{5}}&{}\approx{}&0.600000000000&{}:{}&0.000000000000&{}:{}&0.400000000000&,\\C^*_{\mathbf{5}}&{}\approx{}&1.058823529412&{}:{}&-0.058823529412&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5}}&{}\approx{}&0.608695652174&{}:{}&0.032938076416&{}:{}&0.358366271410&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.471910112360&{}:{}&0.005107252298&{}:{}&0.522982635342&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.646153846154&{}:{}&0.174825174825&{}:{}&0.179020979021&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.760180995475&{}:{}&0.002056766763&{}:{}&0.237762237762&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.090909090909&{}:{}&1.090909090909&,\\B^*_{\mathbf{5}}&{}\approx{}&0.600000000000&{}:{}&0.000000000000&{}:{}&0.400000000000&,\\C^*_{\mathbf{5}}&{}\approx{}&1.058823529412&{}:{}&-0.058823529412&{}:{}&0.000000000000&. \end{alignedat} \]
5 (202)