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”.

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

[Top] > Martin's solution > 0a (011)

0a (011)

Malfatti circles

0a (011)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&, \\ P_{\mathbf{a}}&{}\approx{}&-0.385664306768&{}:{}&0.701047554412&{}:{}&0.684616752356&, \\ P^-_{\mathbf{a}}&{}\approx{}&-0.289829859782&{}:{}&0.671944735269&{}:{}&0.617885124513&, \\ P^+_{\mathbf{a}}&{}\approx{}&-0.448462187365&{}:{}&0.720117893096&{}:{}&0.728344294268&, \\ Q_{\mathbf{a}}&{}\approx{}&-0.253164556962&{}:{}&0.683544303797&{}:{}&0.569620253165&, \\ I^\prime_{\mathbf{a}}&{}\approx{}&-0.528301886792&{}:{}&0.740994854202&{}:{}&0.787307032590&, \end{alignedat} \]
\(I_{\mathbf{a}}\) Incenter
\(P_{\mathbf{a}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{a}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{a}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{a}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{a}}\) Radical center of the Malfatti circles
0a (011)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{a}}&{}\approx{}&-1.700000000000&{}:{}&1.252319109462&{}:{}&1.447680890538&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.273437500000&{}:{}&0.931344696970&{}:{}&0.342092803030&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.311111111111&{}:{}&0.336700336700&{}:{}&0.974410774411&, \\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.291060291060&{}:{}&0.653184653185&{}:{}&0.637875637876&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.605405405405&{}:{}&0.530712530713&{}:{}&1.074692874693&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.546107036979&{}:{}&0.992694931843&{}:{}&0.553412105136&, \\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.180567497850&{}:{}&0.615023841163&{}:{}&0.565543656687&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.535175072278&{}:{}&0.394241158777&{}:{}&1.140933913501&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.460832042487&{}:{}&1.068398076807&{}:{}&0.392433965680&, \\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.545454545455&{}:{}&0.454545454545&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.800000000000&{}:{}&0.000000000000&{}:{}&1.800000000000&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.588235294118&{}:{}&1.588235294118&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{a}}}{B^\prime_{\mathbf{a}}}{C^\prime_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{a}}}{B^{\prime\prime}_{\mathbf{a}}}{C^{\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{a}}}{B^{\prime\prime\prime}_{\mathbf{a}}}{C^{\prime\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^*_{\mathbf{a}}}{B^*_{\mathbf{a}}}{C^*_{\mathbf{a}}}\)
0a (011)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{a}}}}&{}\approx{}&1.542857142857&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{a}}}}&{}\approx{}&0.364583333333&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{a}}}}&{}\approx{}&0.414814814815&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-1.700000000000&{}:{}&1.252319109462&{}:{}&1.447680890538&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.273437500000&{}:{}&0.931344696970&{}:{}&0.342092803030&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.311111111111&{}:{}&0.336700336700&{}:{}&0.974410774411&. \end{alignedat} \]
0a (011)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{a}}&{}\approx{}&-0.385664306768&{}:{}&0.701047554412&{}:{}&0.684616752356&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.291060291060&{}:{}&0.653184653185&{}:{}&0.637875637876&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.605405405405&{}:{}&0.530712530713&{}:{}&1.074692874693&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.546107036979&{}:{}&0.992694931843&{}:{}&0.553412105136&. \end{alignedat} \]
0a (011)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{a}}&{}\approx{}&-0.289829859782&{}:{}&0.671944735269&{}:{}&0.617885124513&,\\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.180567497850&{}:{}&0.615023841163&{}:{}&0.565543656687&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.535175072278&{}:{}&0.394241158777&{}:{}&1.140933913501&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.460832042487&{}:{}&1.068398076807&{}:{}&0.392433965680&. \end{alignedat} \]
0a (011)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{a}}&{}\approx{}&-0.448462187365&{}:{}&0.720117893096&{}:{}&0.728344294268&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-1.700000000000&{}:{}&1.252319109462&{}:{}&1.447680890538&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.273437500000&{}:{}&0.931344696970&{}:{}&0.342092803030&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.311111111111&{}:{}&0.336700336700&{}:{}&0.974410774411&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.291060291060&{}:{}&0.653184653185&{}:{}&0.637875637876&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.605405405405&{}:{}&0.530712530713&{}:{}&1.074692874693&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.546107036979&{}:{}&0.992694931843&{}:{}&0.553412105136&, \end{alignedat} \]
0a (011)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{a}}&{}\approx{}&-0.253164556962&{}:{}&0.683544303797&{}:{}&0.569620253165&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.545454545455&{}:{}&0.454545454545&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.800000000000&{}:{}&0.000000000000&{}:{}&1.800000000000&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.588235294118&{}:{}&1.588235294118&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{a}}&{}\approx{}&-0.528301886792&{}:{}&0.740994854202&{}:{}&0.787307032590&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.291060291060&{}:{}&0.653184653185&{}:{}&0.637875637876&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.605405405405&{}:{}&0.530712530713&{}:{}&1.074692874693&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.546107036979&{}:{}&0.992694931843&{}:{}&0.553412105136&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.545454545455&{}:{}&0.454545454545&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.800000000000&{}:{}&0.000000000000&{}:{}&1.800000000000&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.588235294118&{}:{}&1.588235294118&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)