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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4c (310)

Malfatti circles

4c (310)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&, \\ P_{\mathbf{4c}}&{}\approx{}&0.124342105263&{}:{}&0.972953216374&{}:{}&-0.097295321637&, \\ P^-_{\mathbf{4c}}&{}\approx{}&0.034101711027&{}:{}&0.945421419518&{}:{}&0.020476869455&, \\ P^+_{\mathbf{4c}}&{}\approx{}&0.201649022801&{}:{}&0.996539087948&{}:{}&-0.198188110749&, \\ Q_{\mathbf{4c}}&{}\approx{}&-0.466666666667&{}:{}&1.222222222222&{}:{}&0.244444444444&, \\ I^\prime_{\mathbf{4c}}&{}\approx{}&0.342391304348&{}:{}&0.996376811594&{}:{}&-0.338768115942&, \end{alignedat} \]
\(I_{\mathbf{c}}\) Incenter
\(P_{\mathbf{4c}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{4c}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{4c}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{4c}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{4c}}\) Radical center of the Malfatti circles
4c (310)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4c}}&{}\approx{}&1.993055555556&{}:{}&6.365740740741&{}:{}&-7.358796296296&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.109375000000&{}:{}&1.027462121212&{}:{}&-0.136837121212&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.437500000000&{}:{}&0.473484848485&{}:{}&0.089015151515&, \\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.175000000000&{}:{}&0.916666666667&{}:{}&-0.091666666667&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.545192307692&{}:{}&0.881410256410&{}:{}&-0.426602564103&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.143762677485&{}:{}&1.124915483435&{}:{}&-0.268678160920&, \\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.094362745098&{}:{}&0.886437908497&{}:{}&0.019199346405&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.208575581395&{}:{}&0.666182170543&{}:{}&0.125242248062&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.039949888641&{}:{}&1.107553823311&{}:{}&-0.147503711952&, \\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.833333333333&{}:{}&0.166666666667&,\\B^*_{\mathbf{4c}}&{}\approx{}&2.100000000000&{}:{}&0.000000000000&{}:{}&-1.100000000000&,\\C^*_{\mathbf{4c}}&{}\approx{}&-0.617647058824&{}:{}&1.617647058824&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4c}}}{B^\prime_{\mathbf{4c}}}{C^\prime_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^*_{\mathbf{4c}}}{B^*_{\mathbf{4c}}}{C^*_{\mathbf{4c}}}\)
4c (310)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4c}}}}&{}\approx{}&4.888888888889&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4c}}}}&{}\approx{}&0.090909090909&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4c}}}}&{}\approx{}&0.363636363636&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&1.993055555556&{}:{}&6.365740740741&{}:{}&-7.358796296296&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.109375000000&{}:{}&1.027462121212&{}:{}&-0.136837121212&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.437500000000&{}:{}&0.473484848485&{}:{}&0.089015151515&. \end{alignedat} \]
4c (310)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4c}}&{}\approx{}&0.124342105263&{}:{}&0.972953216374&{}:{}&-0.097295321637&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.175000000000&{}:{}&0.916666666667&{}:{}&-0.091666666667&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.545192307692&{}:{}&0.881410256410&{}:{}&-0.426602564103&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.143762677485&{}:{}&1.124915483435&{}:{}&-0.268678160920&. \end{alignedat} \]
4c (310)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4c}}&{}\approx{}&0.034101711027&{}:{}&0.945421419518&{}:{}&0.020476869455&,\\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.094362745098&{}:{}&0.886437908497&{}:{}&0.019199346405&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.208575581395&{}:{}&0.666182170543&{}:{}&0.125242248062&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.039949888641&{}:{}&1.107553823311&{}:{}&-0.147503711952&. \end{alignedat} \]
4c (310)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4c}}&{}\approx{}&0.201649022801&{}:{}&0.996539087948&{}:{}&-0.198188110749&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&1.993055555556&{}:{}&6.365740740741&{}:{}&-7.358796296296&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.109375000000&{}:{}&1.027462121212&{}:{}&-0.136837121212&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.437500000000&{}:{}&0.473484848485&{}:{}&0.089015151515&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.175000000000&{}:{}&0.916666666667&{}:{}&-0.091666666667&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.545192307692&{}:{}&0.881410256410&{}:{}&-0.426602564103&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.143762677485&{}:{}&1.124915483435&{}:{}&-0.268678160920&, \end{alignedat} \]
4c (310)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4c}}&{}\approx{}&-0.466666666667&{}:{}&1.222222222222&{}:{}&0.244444444444&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.833333333333&{}:{}&0.166666666667&,\\B^*_{\mathbf{4c}}&{}\approx{}&2.100000000000&{}:{}&0.000000000000&{}:{}&-1.100000000000&,\\C^*_{\mathbf{4c}}&{}\approx{}&-0.617647058824&{}:{}&1.617647058824&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4c}}&{}\approx{}&0.342391304348&{}:{}&0.996376811594&{}:{}&-0.338768115942&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.175000000000&{}:{}&0.916666666667&{}:{}&-0.091666666667&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.545192307692&{}:{}&0.881410256410&{}:{}&-0.426602564103&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.143762677485&{}:{}&1.124915483435&{}:{}&-0.268678160920&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.833333333333&{}:{}&0.166666666667&,\\B^*_{\mathbf{4c}}&{}\approx{}&2.100000000000&{}:{}&0.000000000000&{}:{}&-1.100000000000&,\\C^*_{\mathbf{4c}}&{}\approx{}&-0.617647058824&{}:{}&1.617647058824&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)