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,
\[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&, \\ P&{}\approx{}&0.284732052578&{}:{}&0.315903510039&{}:{}&0.399364437383&, \\ P^-&{}\approx{}&0.279172413793&{}:{}&0.312709359606&{}:{}&0.408118226601&, \\ P^+&{}\approx{}&0.287948922586&{}:{}&0.317751681678&{}:{}&0.394299395736&, \\ Q&{}\approx{}&0.275862068966&{}:{}&0.310344827586&{}:{}&0.413793103448&, \\ I^\prime&{}\approx{}&0.292358803987&{}:{}&0.320360702421&{}:{}&0.387280493593&, \end{alignedat} \]
\(I\)
\(P\)
\(P^-\)
\(P^+\)
\(Q\)
\(I^\prime\)
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.522727272727&{}:{}&0.221369539551&{}:{}&0.255903187721&,\\B^\prime&{}\approx{}&0.195555555556&{}:{}&0.559788359788&{}:{}&0.244656084656&,\\C^\prime&{}\approx{}&0.171875000000&{}:{}&0.186011904762&{}:{}&0.642113095238&, \\ A^{\prime\prime}&{}\approx{}&0.182952182952&{}:{}&0.360855360855&{}:{}&0.456192456192&,\\B^{\prime\prime}&{}\approx{}&0.332075471698&{}:{}&0.202156334232&{}:{}&0.465768194070&,\\C^{\prime\prime}&{}\approx{}&0.355465791467&{}:{}&0.394380928337&{}:{}&0.250153280196&, \\ A^{\prime\prime\prime}&{}\approx{}&0.131605184447&{}:{}&0.376726961971&{}:{}&0.491667853582&,\\B^{\prime\prime\prime}&{}\approx{}&0.346575342466&{}:{}&0.146771037182&{}:{}&0.506653620352&,\\C^{\prime\prime\prime}&{}\approx{}&0.383188186293&{}:{}&0.429220533348&{}:{}&0.187591280359&, \\ A^*&{}\approx{}&0.000000000000&{}:{}&0.428571428571&{}:{}&0.571428571429&,\\B^*&{}\approx{}&0.400000000000&{}:{}&0.000000000000&{}:{}&0.600000000000&,\\C^*&{}\approx{}&0.470588235294&{}:{}&0.529411764706&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime}{B^\prime}{C^\prime}\)
\(\triangle{A^{\prime\prime}}{B^{\prime\prime}}{C^{\prime\prime}}\)
\(\triangle{A^{\prime\prime\prime}}{B^{\prime\prime\prime}}{C^{\prime\prime\prime}}\)
\(\triangle{A^*}{B^*}{C^*}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime}}&{}\approx{}&0.681818181818&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime}}&{}\approx{}&0.651851851852&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime}}&{}\approx{}&0.572916666667&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&,\\ A^\prime&{}\approx{}&0.522727272727&{}:{}&0.221369539551&{}:{}&0.255903187721&,\\B^\prime&{}\approx{}&0.195555555556&{}:{}&0.559788359788&{}:{}&0.244656084656&,\\C^\prime&{}\approx{}&0.171875000000&{}:{}&0.186011904762&{}:{}&0.642113095238&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P&{}\approx{}&0.284732052578&{}:{}&0.315903510039&{}:{}&0.399364437383&,\\ A^{\prime\prime}&{}\approx{}&0.182952182952&{}:{}&0.360855360855&{}:{}&0.456192456192&,\\B^{\prime\prime}&{}\approx{}&0.332075471698&{}:{}&0.202156334232&{}:{}&0.465768194070&,\\C^{\prime\prime}&{}\approx{}&0.355465791467&{}:{}&0.394380928337&{}:{}&0.250153280196&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-&{}\approx{}&0.279172413793&{}:{}&0.312709359606&{}:{}&0.408118226601&,\\ A^{\prime\prime\prime}&{}\approx{}&0.131605184447&{}:{}&0.376726961971&{}:{}&0.491667853582&,\\B^{\prime\prime\prime}&{}\approx{}&0.346575342466&{}:{}&0.146771037182&{}:{}&0.506653620352&,\\C^{\prime\prime\prime}&{}\approx{}&0.383188186293&{}:{}&0.429220533348&{}:{}&0.187591280359&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+&{}\approx{}&0.287948922586&{}:{}&0.317751681678&{}:{}&0.394299395736&,\\ A^\prime&{}\approx{}&0.522727272727&{}:{}&0.221369539551&{}:{}&0.255903187721&,\\B^\prime&{}\approx{}&0.195555555556&{}:{}&0.559788359788&{}:{}&0.244656084656&,\\C^\prime&{}\approx{}&0.171875000000&{}:{}&0.186011904762&{}:{}&0.642113095238&,\\ A^{\prime\prime}&{}\approx{}&0.182952182952&{}:{}&0.360855360855&{}:{}&0.456192456192&,\\B^{\prime\prime}&{}\approx{}&0.332075471698&{}:{}&0.202156334232&{}:{}&0.465768194070&,\\C^{\prime\prime}&{}\approx{}&0.355465791467&{}:{}&0.394380928337&{}:{}&0.250153280196&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q&{}\approx{}&0.275862068966&{}:{}&0.310344827586&{}:{}&0.413793103448&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.428571428571&{}:{}&0.571428571429&,\\B^*&{}\approx{}&0.400000000000&{}:{}&0.000000000000&{}:{}&0.600000000000&,\\C^*&{}\approx{}&0.470588235294&{}:{}&0.529411764706&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.292358803987&{}:{}&0.320360702421&{}:{}&0.387280493593&,\\ A^{\prime\prime}&{}\approx{}&0.182952182952&{}:{}&0.360855360855&{}:{}&0.456192456192&,\\B^{\prime\prime}&{}\approx{}&0.332075471698&{}:{}&0.202156334232&{}:{}&0.465768194070&,\\C^{\prime\prime}&{}\approx{}&0.355465791467&{}:{}&0.394380928337&{}:{}&0.250153280196&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.428571428571&{}:{}&0.571428571429&,\\B^*&{}\approx{}&0.400000000000&{}:{}&0.000000000000&{}:{}&0.600000000000&,\\C^*&{}\approx{}&0.470588235294&{}:{}&0.529411764706&{}:{}&0.000000000000&. \end{alignedat} \]
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