Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 108° Top Angle

\(A=36\degree\), \(B=36\degree\), \(C=108\degree\).

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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.276393202250&{}:{}&0.276393202250&{}:{}&0.447213595500&, \\ P_{\mathbf{4}}&{}\approx{}&0.997837970864&{}:{}&0.000627928339&{}:{}&0.001534100797&, \\ P^-_{\mathbf{4}}&{}\approx{}&1.019016937363&{}:{}&-0.007467526437&{}:{}&-0.011549410926&, \\ P^+_{\mathbf{4}}&{}\approx{}&0.977833518729&{}:{}&0.008274436377&{}:{}&0.013892044894&, \\ Q_{\mathbf{4}}&{}\approx{}&1.118301208960&{}:{}&-0.028053291404&{}:{}&-0.090247917556&, \\ I^\prime_{\mathbf{4}}&{}\approx{}&0.930265846940&{}:{}&0.023336305710&{}:{}&0.046397847350&, \end{alignedat} \]
\(I\) Incenter
\(P_{\mathbf{4}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{4}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{4}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{4}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{4}}\) Radical center of the Malfatti circles
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4}}&{}\approx{}&0.979579673804&{}:{}&0.007799870546&{}:{}&0.012620455651&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.589033892950&{}:{}&-0.542110752268&{}:{}&0.953076859319&,\\C^\prime_{\mathbf{4}}&{}\approx{}&1.734342398551&{}:{}&1.734342398551&{}:{}&-2.468684797102&, \\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.879398191730&{}:{}&0.035026953083&{}:{}&0.085574855187&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.982958876324&{}:{}&0.015529898368&{}:{}&0.001511225309&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.974475729089&{}:{}&0.000613226740&{}:{}&0.024911044171&, \\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-0.744185273675&{}:{}&0.684902589423&{}:{}&1.059282684252&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&1.003382813462&{}:{}&0.007989402253&{}:{}&-0.011372215715&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.994475317049&{}:{}&-0.007287681342&{}:{}&0.012812364293&, \\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.237134443940&{}:{}&0.762865556060&,\\B^*_{\mathbf{4}}&{}\approx{}&1.087785252292&{}:{}&0.000000000000&{}:{}&-0.087785252292&,\\C^*_{\mathbf{4}}&{}\approx{}&1.025731112119&{}:{}&-0.025731112119&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4}}}{B^\prime_{\mathbf{4}}}{C^\prime_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4}}}{B^{\prime\prime}_{\mathbf{4}}}{C^{\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4}}}{B^{\prime\prime\prime}_{\mathbf{4}}}{C^{\prime\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^*_{\mathbf{4}}}{B^*_{\mathbf{4}}}{C^*_{\mathbf{4}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4}}}}&{}\approx{}&0.028220196742&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4}}}}&{}\approx{}&2.131144645218&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4}}}}&{}\approx{}&6.274909746087&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.276393202250&{}:{}&0.276393202250&{}:{}&0.447213595500&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.979579673804&{}:{}&0.007799870546&{}:{}&0.012620455651&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.589033892950&{}:{}&-0.542110752268&{}:{}&0.953076859319&,\\C^\prime_{\mathbf{4}}&{}\approx{}&1.734342398551&{}:{}&1.734342398551&{}:{}&-2.468684797102&. \end{alignedat} \]
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First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4}}&{}\approx{}&0.997837970864&{}:{}&0.000627928339&{}:{}&0.001534100797&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.879398191730&{}:{}&0.035026953083&{}:{}&0.085574855187&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.982958876324&{}:{}&0.015529898368&{}:{}&0.001511225309&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.974475729089&{}:{}&0.000613226740&{}:{}&0.024911044171&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4}}&{}\approx{}&1.019016937363&{}:{}&-0.007467526437&{}:{}&-0.011549410926&,\\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-0.744185273675&{}:{}&0.684902589423&{}:{}&1.059282684252&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&1.003382813462&{}:{}&0.007989402253&{}:{}&-0.011372215715&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.994475317049&{}:{}&-0.007287681342&{}:{}&0.012812364293&. \end{alignedat} \]
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Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4}}&{}\approx{}&0.977833518729&{}:{}&0.008274436377&{}:{}&0.013892044894&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.979579673804&{}:{}&0.007799870546&{}:{}&0.012620455651&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.589033892950&{}:{}&-0.542110752268&{}:{}&0.953076859319&,\\C^\prime_{\mathbf{4}}&{}\approx{}&1.734342398551&{}:{}&1.734342398551&{}:{}&-2.468684797102&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.879398191730&{}:{}&0.035026953083&{}:{}&0.085574855187&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.982958876324&{}:{}&0.015529898368&{}:{}&0.001511225309&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.974475729089&{}:{}&0.000613226740&{}:{}&0.024911044171&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4}}&{}\approx{}&1.118301208960&{}:{}&-0.028053291404&{}:{}&-0.090247917556&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.237134443940&{}:{}&0.762865556060&,\\B^*_{\mathbf{4}}&{}\approx{}&1.087785252292&{}:{}&0.000000000000&{}:{}&-0.087785252292&,\\C^*_{\mathbf{4}}&{}\approx{}&1.025731112119&{}:{}&-0.025731112119&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4}}&{}\approx{}&0.930265846940&{}:{}&0.023336305710&{}:{}&0.046397847350&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.879398191730&{}:{}&0.035026953083&{}:{}&0.085574855187&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.982958876324&{}:{}&0.015529898368&{}:{}&0.001511225309&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.974475729089&{}:{}&0.000613226740&{}:{}&0.024911044171&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.237134443940&{}:{}&0.762865556060&,\\B^*_{\mathbf{4}}&{}\approx{}&1.087785252292&{}:{}&0.000000000000&{}:{}&-0.087785252292&,\\C^*_{\mathbf{4}}&{}\approx{}&1.025731112119&{}:{}&-0.025731112119&{}:{}&0.000000000000&. \end{alignedat} \]
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