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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6a(231)

Malfatti circles

6a (231)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&, \\ P_{\mathbf{6a}}&{}\approx{}&1.040077821012&{}:{}&-0.035575319622&{}:{}&-0.004502501390&, \\ P^-_{\mathbf{6a}}&{}\approx{}&0.912030346821&{}:{}&0.025030966144&{}:{}&0.062938687036&, \\ P^+_{\mathbf{6a}}&{}\approx{}&1.189502529511&{}:{}&-0.106299686822&{}:{}&-0.083202842689&, \\ Q_{\mathbf{6a}}&{}\approx{}&1.350000000000&{}:{}&-0.500000000000&{}:{}&0.150000000000&, \\ I^\prime_{\mathbf{6a}}&{}\approx{}&1.362385321101&{}:{}&-0.262123197903&{}:{}&-0.100262123198&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{6a}}\)
\(P^-_{\mathbf{6a}}\)
\(P^+_{\mathbf{6a}}\)
\(Q_{\mathbf{6a}}\)
\(I^\prime_{\mathbf{6a}}\)
6a (231)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6a}}&{}\approx{}&1.159090909091&{}:{}&-0.073789846517&{}:{}&-0.085301062574&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&4.640625000000&{}:{}&2.165178571429&{}:{}&-5.805803571429&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.916666666667&{}:{}&-0.992063492063&{}:{}&1.075396825397&, \\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.530927835052&{}:{}&-0.471281296024&{}:{}&-0.059646539028&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.142307692308&{}:{}&-0.137362637363&{}:{}&-0.004945054945&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.209502262443&{}:{}&-0.041370394312&{}:{}&-0.168131868132&, \\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.378826530612&{}:{}&0.176749271137&{}:{}&0.444424198251&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.994680851064&{}:{}&-0.063323201621&{}:{}&0.068642350557&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.048380398671&{}:{}&0.028773137162&{}:{}&-0.077153535833&, \\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.428571428571&{}:{}&-0.428571428571&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.900000000000&{}:{}&0.000000000000&{}:{}&0.100000000000&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.588235294118&{}:{}&-0.588235294118&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6a}}}{B^\prime_{\mathbf{6a}}}{C^\prime_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^*_{\mathbf{6a}}}{B^*_{\mathbf{6a}}}{C^*_{\mathbf{6a}}}\)
6a (231)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6a}}}}&{}\approx{}&-0.090909090909&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6a}}}}&{}\approx{}&-6.187500000000&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6a}}}}&{}\approx{}&-1.222222222222&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.159090909091&{}:{}&-0.073789846517&{}:{}&-0.085301062574&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&4.640625000000&{}:{}&2.165178571429&{}:{}&-5.805803571429&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.916666666667&{}:{}&-0.992063492063&{}:{}&1.075396825397&. \end{alignedat} \]
6a (231)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6a}}&{}\approx{}&1.040077821012&{}:{}&-0.035575319622&{}:{}&-0.004502501390&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.530927835052&{}:{}&-0.471281296024&{}:{}&-0.059646539028&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.142307692308&{}:{}&-0.137362637363&{}:{}&-0.004945054945&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.209502262443&{}:{}&-0.041370394312&{}:{}&-0.168131868132&. \end{alignedat} \]
6a (231)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6a}}&{}\approx{}&0.912030346821&{}:{}&0.025030966144&{}:{}&0.062938687036&,\\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.378826530612&{}:{}&0.176749271137&{}:{}&0.444424198251&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.994680851064&{}:{}&-0.063323201621&{}:{}&0.068642350557&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.048380398671&{}:{}&0.028773137162&{}:{}&-0.077153535833&. \end{alignedat} \]
6a (231)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6a}}&{}\approx{}&1.189502529511&{}:{}&-0.106299686822&{}:{}&-0.083202842689&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.159090909091&{}:{}&-0.073789846517&{}:{}&-0.085301062574&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&4.640625000000&{}:{}&2.165178571429&{}:{}&-5.805803571429&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.916666666667&{}:{}&-0.992063492063&{}:{}&1.075396825397&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.530927835052&{}:{}&-0.471281296024&{}:{}&-0.059646539028&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.142307692308&{}:{}&-0.137362637363&{}:{}&-0.004945054945&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.209502262443&{}:{}&-0.041370394312&{}:{}&-0.168131868132&, \end{alignedat} \]
6a (231)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6a}}&{}\approx{}&1.350000000000&{}:{}&-0.500000000000&{}:{}&0.150000000000&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.428571428571&{}:{}&-0.428571428571&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.900000000000&{}:{}&0.000000000000&{}:{}&0.100000000000&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.588235294118&{}:{}&-0.588235294118&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6a}}&{}\approx{}&1.362385321101&{}:{}&-0.262123197903&{}:{}&-0.100262123198&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.530927835052&{}:{}&-0.471281296024&{}:{}&-0.059646539028&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.142307692308&{}:{}&-0.137362637363&{}:{}&-0.004945054945&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.209502262443&{}:{}&-0.041370394312&{}:{}&-0.168131868132&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.428571428571&{}:{}&-0.428571428571&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.900000000000&{}:{}&0.000000000000&{}:{}&0.100000000000&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.588235294118&{}:{}&-0.588235294118&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)