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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4b (301)

Malfatti circles

4b (301)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&, \\ P_{\mathbf{4b}}&{}\approx{}&0.061628406844&{}:{}&-0.038145186714&{}:{}&0.976516779870&, \\ P^-_{\mathbf{4b}}&{}\approx{}&0.003616569794&{}:{}&0.026724482578&{}:{}&0.969658947627&, \\ P^+_{\mathbf{4b}}&{}\approx{}&0.112243419464&{}:{}&-0.094743620704&{}:{}&0.982500201240&, \\ Q_{\mathbf{4b}}&{}\approx{}&-0.394366197183&{}:{}&0.154929577465&{}:{}&1.239436619718&, \\ I^\prime_{\mathbf{4b}}&{}\approx{}&0.215799614644&{}:{}&-0.176621708414&{}:{}&0.960822093770&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{4b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{4b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{4b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{4b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{4b}}\) Radical center of the Malfatti circles
4b (301)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4b}}&{}\approx{}&0.255208333333&{}:{}&-4.774305555556&{}:{}&5.519097222222&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.373333333333&{}:{}&0.159595959596&{}:{}&0.467070707071&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.064814814815&{}:{}&-0.070145903479&{}:{}&1.005331088664&, \\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.110453648915&{}:{}&-0.036160420776&{}:{}&0.925706771861&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.067571644042&{}:{}&-0.138260432378&{}:{}&1.070688788336&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.364116631108&{}:{}&-0.225371668529&{}:{}&0.861255037421&, \\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.059037761165&{}:{}&0.025238003961&{}:{}&0.915724234873&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.003993480033&{}:{}&-0.074707959794&{}:{}&1.070714479761&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.033316335203&{}:{}&0.246189585812&{}:{}&0.720494078985&, \\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.111111111111&{}:{}&0.888888888889&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.466666666667&{}:{}&0.000000000000&{}:{}&1.466666666667&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.647058823529&{}:{}&-0.647058823529&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4b}}}{B^\prime_{\mathbf{4b}}}{C^\prime_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^*_{\mathbf{4b}}}{B^*_{\mathbf{4b}}}{C^*_{\mathbf{4b}}}\)
4b (301)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4b}}}}&{}\approx{}&5.156250000000&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4b}}}}&{}\approx{}&0.436363636364&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4b}}}}&{}\approx{}&0.075757575758&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&0.255208333333&{}:{}&-4.774305555556&{}:{}&5.519097222222&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.373333333333&{}:{}&0.159595959596&{}:{}&0.467070707071&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.064814814815&{}:{}&-0.070145903479&{}:{}&1.005331088664&. \end{alignedat} \]
4b (301)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4b}}&{}\approx{}&0.061628406844&{}:{}&-0.038145186714&{}:{}&0.976516779870&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.110453648915&{}:{}&-0.036160420776&{}:{}&0.925706771861&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.067571644042&{}:{}&-0.138260432378&{}:{}&1.070688788336&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.364116631108&{}:{}&-0.225371668529&{}:{}&0.861255037421&. \end{alignedat} \]
4b (301)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4b}}&{}\approx{}&0.003616569794&{}:{}&0.026724482578&{}:{}&0.969658947627&,\\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.059037761165&{}:{}&0.025238003961&{}:{}&0.915724234873&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.003993480033&{}:{}&-0.074707959794&{}:{}&1.070714479761&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.033316335203&{}:{}&0.246189585812&{}:{}&0.720494078985&. \end{alignedat} \]
4b (301)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4b}}&{}\approx{}&0.112243419464&{}:{}&-0.094743620704&{}:{}&0.982500201240&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&0.255208333333&{}:{}&-4.774305555556&{}:{}&5.519097222222&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.373333333333&{}:{}&0.159595959596&{}:{}&0.467070707071&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.064814814815&{}:{}&-0.070145903479&{}:{}&1.005331088664&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.110453648915&{}:{}&-0.036160420776&{}:{}&0.925706771861&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.067571644042&{}:{}&-0.138260432378&{}:{}&1.070688788336&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.364116631108&{}:{}&-0.225371668529&{}:{}&0.861255037421&, \end{alignedat} \]
4b (301)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4b}}&{}\approx{}&-0.394366197183&{}:{}&0.154929577465&{}:{}&1.239436619718&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.111111111111&{}:{}&0.888888888889&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.466666666667&{}:{}&0.000000000000&{}:{}&1.466666666667&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.647058823529&{}:{}&-0.647058823529&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4b}}&{}\approx{}&0.215799614644&{}:{}&-0.176621708414&{}:{}&0.960822093770&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.110453648915&{}:{}&-0.036160420776&{}:{}&0.925706771861&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.067571644042&{}:{}&-0.138260432378&{}:{}&1.070688788336&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.364116631108&{}:{}&-0.225371668529&{}:{}&0.861255037421&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.111111111111&{}:{}&0.888888888889&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.466666666667&{}:{}&0.000000000000&{}:{}&1.466666666667&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.647058823529&{}:{}&-0.647058823529&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)