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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7c (332)

Malfatti circles

7c (332)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&, \\ P_{\mathbf{7c}}&{}\approx{}&-0.016295441001&{}:{}&-0.016525129122&{}:{}&1.032820570123&, \\ P^-_{\mathbf{7c}}&{}\approx{}&0.022405150029&{}:{}&0.025323379496&{}:{}&0.952271470475&, \\ P^+_{\mathbf{7c}}&{}\approx{}&-0.057618992606&{}:{}&-0.061209950709&{}:{}&1.118828943315&, \\ Q_{\mathbf{7c}}&{}\approx{}&0.150214592275&{}:{}&0.141630901288&{}:{}&0.708154506438&, \\ I^\prime_{\mathbf{7c}}&{}\approx{}&-0.145833333333&{}:{}&-0.152777777778&{}:{}&1.298611111111&, \end{alignedat} \]
\(I_{\mathbf{c}}\) Incenter
\(P_{\mathbf{7c}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{7c}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{7c}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{7c}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{7c}}\) Radical center of the Malfatti circles
7c (332)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7c}}&{}\approx{}&0.776562500000&{}:{}&-1.432291666667&{}:{}&1.655729166667&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-1.093750000000&{}:{}&0.725378787879&{}:{}&1.368371212121&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.043750000000&{}:{}&-0.047348484848&{}:{}&1.091098484848&, \\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.084134615385&{}:{}&-0.017628205128&{}:{}&1.101762820513&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.017500000000&{}:{}&-0.091666666667&{}:{}&1.109166666667&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.247454751131&{}:{}&-0.250942684766&{}:{}&1.498397435897&, \\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.040646114865&{}:{}&0.026956644144&{}:{}&1.013689470721&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.024005023184&{}:{}&-0.044274858320&{}:{}&1.020269835137&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.234611027190&{}:{}&0.265168680765&{}:{}&0.500220292044&, \\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.166666666667&{}:{}&0.833333333333&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.175000000000&{}:{}&0.000000000000&{}:{}&0.825000000000&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.514705882353&{}:{}&0.485294117647&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7c}}}{B^\prime_{\mathbf{7c}}}{C^\prime_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^*_{\mathbf{7c}}}{B^*_{\mathbf{7c}}}{C^*_{\mathbf{7c}}}\)
7c (332)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7c}}}}&{}\approx{}&-1.100000000000&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.909090909091&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.036363636364&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&0.776562500000&{}:{}&-1.432291666667&{}:{}&1.655729166667&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-1.093750000000&{}:{}&0.725378787879&{}:{}&1.368371212121&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.043750000000&{}:{}&-0.047348484848&{}:{}&1.091098484848&. \end{alignedat} \]
7c (332)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7c}}&{}\approx{}&-0.016295441001&{}:{}&-0.016525129122&{}:{}&1.032820570123&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.084134615385&{}:{}&-0.017628205128&{}:{}&1.101762820513&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.017500000000&{}:{}&-0.091666666667&{}:{}&1.109166666667&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.247454751131&{}:{}&-0.250942684766&{}:{}&1.498397435897&. \end{alignedat} \]
7c (332)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7c}}&{}\approx{}&0.022405150029&{}:{}&0.025323379496&{}:{}&0.952271470475&,\\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.040646114865&{}:{}&0.026956644144&{}:{}&1.013689470721&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.024005023184&{}:{}&-0.044274858320&{}:{}&1.020269835137&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.234611027190&{}:{}&0.265168680765&{}:{}&0.500220292044&. \end{alignedat} \]
7c (332)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7c}}&{}\approx{}&-0.057618992606&{}:{}&-0.061209950709&{}:{}&1.118828943315&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&0.776562500000&{}:{}&-1.432291666667&{}:{}&1.655729166667&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-1.093750000000&{}:{}&0.725378787879&{}:{}&1.368371212121&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.043750000000&{}:{}&-0.047348484848&{}:{}&1.091098484848&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.084134615385&{}:{}&-0.017628205128&{}:{}&1.101762820513&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.017500000000&{}:{}&-0.091666666667&{}:{}&1.109166666667&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.247454751131&{}:{}&-0.250942684766&{}:{}&1.498397435897&, \end{alignedat} \]
7c (332)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7c}}&{}\approx{}&0.150214592275&{}:{}&0.141630901288&{}:{}&0.708154506438&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.166666666667&{}:{}&0.833333333333&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.175000000000&{}:{}&0.000000000000&{}:{}&0.825000000000&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.514705882353&{}:{}&0.485294117647&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7c}}&{}\approx{}&-0.145833333333&{}:{}&-0.152777777778&{}:{}&1.298611111111&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.084134615385&{}:{}&-0.017628205128&{}:{}&1.101762820513&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.017500000000&{}:{}&-0.091666666667&{}:{}&1.109166666667&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.247454751131&{}:{}&-0.250942684766&{}:{}&1.498397435897&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.166666666667&{}:{}&0.833333333333&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.175000000000&{}:{}&0.000000000000&{}:{}&0.825000000000&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.514705882353&{}:{}&0.485294117647&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)