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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3b (123)

Malfatti circles

3b (123)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&, \\ P_{\mathbf{3b}}&{}\approx{}&-0.007061790668&{}:{}&1.118957545187&{}:{}&-0.111895754519&, \\ P^-_{\mathbf{3b}}&{}\approx{}&0.088677130045&{}:{}&0.892002989537&{}:{}&0.019319880419&, \\ P^+_{\mathbf{3b}}&{}\approx{}&-0.130115273775&{}:{}&1.410662824207&{}:{}&-0.280547550432&, \\ Q_{\mathbf{3b}}&{}\approx{}&0.241379310345&{}:{}&1.517241379310&{}:{}&-0.758620689655&, \\ I^\prime_{\mathbf{3b}}&{}\approx{}&-0.130841121495&{}:{}&1.713395638629&{}:{}&-0.582554517134&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{3b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{3b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{3b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{3b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{3b}}\) Radical center of the Malfatti circles
3b (123)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3b}}&{}\approx{}&1.220679012346&{}:{}&1.414609053498&{}:{}&-1.635288065844&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.140000000000&{}:{}&1.315151515152&{}:{}&-0.175151515152&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-3.500000000000&{}:{}&3.787878787879&{}:{}&0.712121212121&, \\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.269230769231&{}:{}&1.410256410256&{}:{}&-0.141025641026&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.062921348315&{}:{}&2.059925093633&{}:{}&-0.997003745318&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.008360704688&{}:{}&1.324773564248&{}:{}&-0.316412859560&, \\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.116314199396&{}:{}&1.092648539778&{}:{}&0.023665659617&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.420744680851&{}:{}&0.487588652482&{}:{}&0.091666666667&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.102887617066&{}:{}&1.034946236559&{}:{}&-0.137833853625&, \\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&2.000000000000&{}:{}&-1.000000000000&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.466666666667&{}:{}&0.000000000000&{}:{}&1.466666666667&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.137254901961&{}:{}&0.862745098039&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3b}}}{B^\prime_{\mathbf{3b}}}{C^\prime_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^*_{\mathbf{3b}}}{B^*_{\mathbf{3b}}}{C^*_{\mathbf{3b}}}\)
3b (123)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3b}}}}&{}\approx{}&-1.527777777778&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3b}}}}&{}\approx{}&-0.163636363636&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3b}}}}&{}\approx{}&-4.090909090909&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&1.220679012346&{}:{}&1.414609053498&{}:{}&-1.635288065844&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.140000000000&{}:{}&1.315151515152&{}:{}&-0.175151515152&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-3.500000000000&{}:{}&3.787878787879&{}:{}&0.712121212121&. \end{alignedat} \]
3b (123)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3b}}&{}\approx{}&-0.007061790668&{}:{}&1.118957545187&{}:{}&-0.111895754519&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.269230769231&{}:{}&1.410256410256&{}:{}&-0.141025641026&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.062921348315&{}:{}&2.059925093633&{}:{}&-0.997003745318&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.008360704688&{}:{}&1.324773564248&{}:{}&-0.316412859560&. \end{alignedat} \]
3b (123)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3b}}&{}\approx{}&0.088677130045&{}:{}&0.892002989537&{}:{}&0.019319880419&,\\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.116314199396&{}:{}&1.092648539778&{}:{}&0.023665659617&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.420744680851&{}:{}&0.487588652482&{}:{}&0.091666666667&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.102887617066&{}:{}&1.034946236559&{}:{}&-0.137833853625&. \end{alignedat} \]
3b (123)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3b}}&{}\approx{}&-0.130115273775&{}:{}&1.410662824207&{}:{}&-0.280547550432&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&1.220679012346&{}:{}&1.414609053498&{}:{}&-1.635288065844&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.140000000000&{}:{}&1.315151515152&{}:{}&-0.175151515152&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-3.500000000000&{}:{}&3.787878787879&{}:{}&0.712121212121&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.269230769231&{}:{}&1.410256410256&{}:{}&-0.141025641026&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.062921348315&{}:{}&2.059925093633&{}:{}&-0.997003745318&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.008360704688&{}:{}&1.324773564248&{}:{}&-0.316412859560&, \end{alignedat} \]
3b (123)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3b}}&{}\approx{}&0.241379310345&{}:{}&1.517241379310&{}:{}&-0.758620689655&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&2.000000000000&{}:{}&-1.000000000000&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.466666666667&{}:{}&0.000000000000&{}:{}&1.466666666667&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.137254901961&{}:{}&0.862745098039&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3b}}&{}\approx{}&-0.130841121495&{}:{}&1.713395638629&{}:{}&-0.582554517134&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.269230769231&{}:{}&1.410256410256&{}:{}&-0.141025641026&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.062921348315&{}:{}&2.059925093633&{}:{}&-0.997003745318&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.008360704688&{}:{}&1.324773564248&{}:{}&-0.316412859560&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&2.000000000000&{}:{}&-1.000000000000&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.466666666667&{}:{}&0.000000000000&{}:{}&1.466666666667&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.137254901961&{}:{}&0.862745098039&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)