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,
\[ \begin{alignedat}{4} I&{}\approx{}&0.276393202250&{}:{}&0.276393202250&{}:{}&0.447213595500&, \\ P_{\mathbf{1}}&{}\approx{}&0.026145372443&{}:{}&0.026145372443&{}:{}&0.947709255114&, \\ P^-_{\mathbf{1}}&{}\approx{}&-0.006289583863&{}:{}&-0.006289583863&{}:{}&1.012579167725&, \\ P^+_{\mathbf{1}}&{}\approx{}&0.051903291889&{}:{}&0.051903291889&{}:{}&0.896193416221&, \\ Q_{\mathbf{1}}&{}\approx{}&-0.096233099183&{}:{}&-0.096233099183&{}:{}&1.192466198366&, \\ I^\prime_{\mathbf{1}}&{}\approx{}&0.103537589192&{}:{}&0.103537589192&{}:{}&0.792924821616&, \end{alignedat} \]
\(I\) Incenter
\(P_{\mathbf{1}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{1}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{1}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{1}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{1}}\) Radical center of the Malfatti circles
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1}}&{}\approx{}&-0.071882550435&{}:{}&0.409422702318&{}:{}&0.662459848116&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.409422702318&{}:{}&-0.071882550435&{}:{}&0.662459848116&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.033040781847&{}:{}&0.033040781847&{}:{}&0.933918436306&, \\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.061146949634&{}:{}&0.025205674416&{}:{}&0.913647375950&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.025205674416&{}:{}&0.061146949634&{}:{}&0.913647375950&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.168770075942&{}:{}&0.168770075942&{}:{}&0.662459848116&, \\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.034376006746&{}:{}&-0.006035412850&{}:{}&0.971659406104&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.006035412850&{}:{}&0.034376006746&{}:{}&0.971659406104&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.138583532638&{}:{}&-0.138583532638&{}:{}&1.277167065277&, \\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.087785252292&{}:{}&1.087785252292&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.087785252292&{}:{}&0.000000000000&{}:{}&1.087785252292&,\\C^*_{\mathbf{1}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1}}}{B^\prime_{\mathbf{1}}}{C^\prime_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1}}}{B^{\prime\prime}_{\mathbf{1}}}{C^{\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1}}}{B^{\prime\prime\prime}_{\mathbf{1}}}{C^{\prime\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^*_{\mathbf{1}}}{B^*_{\mathbf{1}}}{C^*_{\mathbf{1}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1}}}}&{}\approx{}&1.481305252753&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1}}}}&{}\approx{}&1.481305252753&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1}}}}&{}\approx{}&0.119542671737&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.276393202250&{}:{}&0.276393202250&{}:{}&0.447213595500&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.071882550435&{}:{}&0.409422702318&{}:{}&0.662459848116&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.409422702318&{}:{}&-0.071882550435&{}:{}&0.662459848116&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.033040781847&{}:{}&0.033040781847&{}:{}&0.933918436306&. \end{alignedat} \]
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First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1}}&{}\approx{}&0.026145372443&{}:{}&0.026145372443&{}:{}&0.947709255114&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.061146949634&{}:{}&0.025205674416&{}:{}&0.913647375950&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.025205674416&{}:{}&0.061146949634&{}:{}&0.913647375950&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.168770075942&{}:{}&0.168770075942&{}:{}&0.662459848116&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1}}&{}\approx{}&-0.006289583863&{}:{}&-0.006289583863&{}:{}&1.012579167725&,\\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.034376006746&{}:{}&-0.006035412850&{}:{}&0.971659406104&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.006035412850&{}:{}&0.034376006746&{}:{}&0.971659406104&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.138583532638&{}:{}&-0.138583532638&{}:{}&1.277167065277&. \end{alignedat} \]
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Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1}}&{}\approx{}&0.051903291889&{}:{}&0.051903291889&{}:{}&0.896193416221&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.071882550435&{}:{}&0.409422702318&{}:{}&0.662459848116&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.409422702318&{}:{}&-0.071882550435&{}:{}&0.662459848116&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.033040781847&{}:{}&0.033040781847&{}:{}&0.933918436306&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.061146949634&{}:{}&0.025205674416&{}:{}&0.913647375950&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.025205674416&{}:{}&0.061146949634&{}:{}&0.913647375950&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.168770075942&{}:{}&0.168770075942&{}:{}&0.662459848116&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1}}&{}\approx{}&-0.096233099183&{}:{}&-0.096233099183&{}:{}&1.192466198366&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.087785252292&{}:{}&1.087785252292&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.087785252292&{}:{}&0.000000000000&{}:{}&1.087785252292&,\\C^*_{\mathbf{1}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1}}&{}\approx{}&0.103537589192&{}:{}&0.103537589192&{}:{}&0.792924821616&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.061146949634&{}:{}&0.025205674416&{}:{}&0.913647375950&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.025205674416&{}:{}&0.061146949634&{}:{}&0.913647375950&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.168770075942&{}:{}&0.168770075942&{}:{}&0.662459848116&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.087785252292&{}:{}&1.087785252292&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.087785252292&{}:{}&0.000000000000&{}:{}&1.087785252292&,\\C^*_{\mathbf{1}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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