Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 135° Top Angle

\(A=22.5\degree\), \(B=22.5\degree\), \(C=135\degree\).

[Top] > Isosceles Triangle with 135° Top Angle > 4 (200)

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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.259891532474&{}:{}&0.259891532474&{}:{}&0.480216935052&, \\ P_{\mathbf{4}}&{}\approx{}&0.999549267060&{}:{}&0.000094058383&{}:{}&0.000356674557&, \\ P^-_{\mathbf{4}}&{}\approx{}&1.009664732823&{}:{}&-0.003458899062&{}:{}&-0.006205833762&, \\ P^+_{\mathbf{4}}&{}\approx{}&0.989703110874&{}:{}&0.003552423498&{}:{}&0.006744465628&, \\ Q_{\mathbf{4}}&{}\approx{}&1.081677943109&{}:{}&-0.010492878040&{}:{}&-0.071185065069&, \\ I^\prime_{\mathbf{4}}&{}\approx{}&0.965830569058&{}:{}&0.009369094039&{}:{}&0.024800336903&, \end{alignedat} \]
\(I\) 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,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4}}&{}\approx{}&0.990074754576&{}:{}&0.003485282707&{}:{}&0.006439962716&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.533451499915&{}:{}&-0.519141344633&{}:{}&0.985689844718&,\\C^\prime_{\mathbf{4}}&{}\approx{}&3.937549278574&{}:{}&3.937549278574&{}:{}&-6.875098557148&, \\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.939606890140&{}:{}&0.012602758174&{}:{}&0.047790351686&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.992681375206&{}:{}&0.006964400944&{}:{}&0.000354223850&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.987110790876&{}:{}&0.000092887912&{}:{}&0.012796321212&, \\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-0.581662735530&{}:{}&0.566059285041&{}:{}&1.015603450489&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&1.002633123515&{}:{}&0.003529490951&{}:{}&-0.006162614466&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.996930717527&{}:{}&-0.003415275003&{}:{}&0.006484557476&, \\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.128466482391&{}:{}&0.871533517609&,\\B^*_{\mathbf{4}}&{}\approx{}&1.070445884989&{}:{}&0.000000000000&{}:{}&-0.070445884989&,\\C^*_{\mathbf{4}}&{}\approx{}&1.009795579104&{}:{}&-0.009795579104&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4}}}{B^\prime_{\mathbf{4}}}{C^\prime_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4}}}{B^{\prime\prime}_{\mathbf{4}}}{C^{\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4}}}{B^{\prime\prime\prime}_{\mathbf{4}}}{C^{\prime\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^*_{\mathbf{4}}}{B^*_{\mathbf{4}}}{C^*_{\mathbf{4}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4}}}}&{}\approx{}&0.013410528131&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4}}}}&{}\approx{}&2.052592844548&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4}}}}&{}\approx{}&15.150740930607&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.259891532474&{}:{}&0.259891532474&{}:{}&0.480216935052&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.990074754576&{}:{}&0.003485282707&{}:{}&0.006439962716&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.533451499915&{}:{}&-0.519141344633&{}:{}&0.985689844718&,\\C^\prime_{\mathbf{4}}&{}\approx{}&3.937549278574&{}:{}&3.937549278574&{}:{}&-6.875098557148&. \end{alignedat} \]
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First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4}}&{}\approx{}&0.999549267060&{}:{}&0.000094058383&{}:{}&0.000356674557&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.939606890140&{}:{}&0.012602758174&{}:{}&0.047790351686&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.992681375206&{}:{}&0.006964400944&{}:{}&0.000354223850&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.987110790876&{}:{}&0.000092887912&{}:{}&0.012796321212&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4}}&{}\approx{}&1.009664732823&{}:{}&-0.003458899062&{}:{}&-0.006205833762&,\\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-0.581662735530&{}:{}&0.566059285041&{}:{}&1.015603450489&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&1.002633123515&{}:{}&0.003529490951&{}:{}&-0.006162614466&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.996930717527&{}:{}&-0.003415275003&{}:{}&0.006484557476&. \end{alignedat} \]
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Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4}}&{}\approx{}&0.989703110874&{}:{}&0.003552423498&{}:{}&0.006744465628&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.990074754576&{}:{}&0.003485282707&{}:{}&0.006439962716&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.533451499915&{}:{}&-0.519141344633&{}:{}&0.985689844718&,\\C^\prime_{\mathbf{4}}&{}\approx{}&3.937549278574&{}:{}&3.937549278574&{}:{}&-6.875098557148&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.939606890140&{}:{}&0.012602758174&{}:{}&0.047790351686&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.992681375206&{}:{}&0.006964400944&{}:{}&0.000354223850&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.987110790876&{}:{}&0.000092887912&{}:{}&0.012796321212&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4}}&{}\approx{}&1.081677943109&{}:{}&-0.010492878040&{}:{}&-0.071185065069&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.128466482391&{}:{}&0.871533517609&,\\B^*_{\mathbf{4}}&{}\approx{}&1.070445884989&{}:{}&0.000000000000&{}:{}&-0.070445884989&,\\C^*_{\mathbf{4}}&{}\approx{}&1.009795579104&{}:{}&-0.009795579104&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4}}&{}\approx{}&0.965830569058&{}:{}&0.009369094039&{}:{}&0.024800336903&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.939606890140&{}:{}&0.012602758174&{}:{}&0.047790351686&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.992681375206&{}:{}&0.006964400944&{}:{}&0.000354223850&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.987110790876&{}:{}&0.000092887912&{}:{}&0.012796321212&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.128466482391&{}:{}&0.871533517609&,\\B^*_{\mathbf{4}}&{}\approx{}&1.070445884989&{}:{}&0.000000000000&{}:{}&-0.070445884989&,\\C^*_{\mathbf{4}}&{}\approx{}&1.009795579104&{}:{}&-0.009795579104&{}:{}&0.000000000000&. \end{alignedat} \]
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