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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3c(132)

Malfatti circles

3c (132)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&, \\ P_{\mathbf{3c}}&{}\approx{}&-0.018329741379&{}:{}&-0.143426724138&{}:{}&1.161756465517&, \\ P^-_{\mathbf{3c}}&{}\approx{}&0.095142907413&{}:{}&-0.009139427588&{}:{}&0.913996520175&, \\ P^+_{\mathbf{3c}}&{}\approx{}&-0.157696648098&{}:{}&-0.308358144449&{}:{}&1.466054792547&, \\ Q_{\mathbf{3c}}&{}\approx{}&0.323076923077&{}:{}&-0.846153846154&{}:{}&1.523076923077&, \\ I^\prime_{\mathbf{3c}}&{}\approx{}&-0.200211864407&{}:{}&-0.582627118644&{}:{}&1.782838983051&, \end{alignedat} \]
\(I_{\mathbf{c}}\) Incenter
\(P_{\mathbf{3c}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{3c}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{3c}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{3c}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{3c}}\) Radical center of the Malfatti circles
3c (132)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3c}}&{}\approx{}&0.834490740741&{}:{}&-1.060956790123&{}:{}&1.226466049383&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-5.906250000000&{}:{}&-0.482954545455&{}:{}&7.389204545455&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.164062500000&{}:{}&-0.177556818182&{}:{}&1.341619318182&, \\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.319256756757&{}:{}&-0.185810810811&{}:{}&1.505067567568&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.020907079646&{}:{}&-0.304203539823&{}:{}&1.325110619469&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.125073529412&{}:{}&-0.978676470588&{}:{}&2.103750000000&, \\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.140930043384&{}:{}&-0.011523861171&{}:{}&1.152453904555&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.107121632330&{}:{}&-0.136192551506&{}:{}&1.029070919176&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.421290685596&{}:{}&-0.040469182825&{}:{}&0.619178497230&, \\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-1.250000000000&{}:{}&2.250000000000&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.175000000000&{}:{}&0.000000000000&{}:{}&0.825000000000&,\\C^*_{\mathbf{3c}}&{}\approx{}&-0.617647058824&{}:{}&1.617647058824&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3c}}}{B^\prime_{\mathbf{3c}}}{C^\prime_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^*_{\mathbf{3c}}}{B^*_{\mathbf{3c}}}{C^*_{\mathbf{3c}}}\)
3c (132)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.814814814815&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3c}}}}&{}\approx{}&-4.909090909091&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.136363636364&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.834490740741&{}:{}&-1.060956790123&{}:{}&1.226466049383&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-5.906250000000&{}:{}&-0.482954545455&{}:{}&7.389204545455&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.164062500000&{}:{}&-0.177556818182&{}:{}&1.341619318182&. \end{alignedat} \]
3c (132)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3c}}&{}\approx{}&-0.018329741379&{}:{}&-0.143426724138&{}:{}&1.161756465517&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.319256756757&{}:{}&-0.185810810811&{}:{}&1.505067567568&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.020907079646&{}:{}&-0.304203539823&{}:{}&1.325110619469&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.125073529412&{}:{}&-0.978676470588&{}:{}&2.103750000000&. \end{alignedat} \]
3c (132)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3c}}&{}\approx{}&0.095142907413&{}:{}&-0.009139427588&{}:{}&0.913996520175&,\\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.140930043384&{}:{}&-0.011523861171&{}:{}&1.152453904555&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.107121632330&{}:{}&-0.136192551506&{}:{}&1.029070919176&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.421290685596&{}:{}&-0.040469182825&{}:{}&0.619178497230&. \end{alignedat} \]
3c (132)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3c}}&{}\approx{}&-0.157696648098&{}:{}&-0.308358144449&{}:{}&1.466054792547&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.834490740741&{}:{}&-1.060956790123&{}:{}&1.226466049383&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-5.906250000000&{}:{}&-0.482954545455&{}:{}&7.389204545455&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.164062500000&{}:{}&-0.177556818182&{}:{}&1.341619318182&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.319256756757&{}:{}&-0.185810810811&{}:{}&1.505067567568&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.020907079646&{}:{}&-0.304203539823&{}:{}&1.325110619469&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.125073529412&{}:{}&-0.978676470588&{}:{}&2.103750000000&, \end{alignedat} \]
3c (132)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3c}}&{}\approx{}&0.323076923077&{}:{}&-0.846153846154&{}:{}&1.523076923077&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-1.250000000000&{}:{}&2.250000000000&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.175000000000&{}:{}&0.000000000000&{}:{}&0.825000000000&,\\C^*_{\mathbf{3c}}&{}\approx{}&-0.617647058824&{}:{}&1.617647058824&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3c}}&{}\approx{}&-0.200211864407&{}:{}&-0.582627118644&{}:{}&1.782838983051&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.319256756757&{}:{}&-0.185810810811&{}:{}&1.505067567568&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.020907079646&{}:{}&-0.304203539823&{}:{}&1.325110619469&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.125073529412&{}:{}&-0.978676470588&{}:{}&2.103750000000&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-1.250000000000&{}:{}&2.250000000000&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.175000000000&{}:{}&0.000000000000&{}:{}&0.825000000000&,\\C^*_{\mathbf{3c}}&{}\approx{}&-0.617647058824&{}:{}&1.617647058824&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)