Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 135° Top Angle

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


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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.259891532474&{}:{}&0.259891532474&{}:{}&0.480216935052&, \\ P_{\mathbf{1}}&{}\approx{}&0.047564446284&{}:{}&0.047564446284&{}:{}&0.904871107431&, \\ P^-_{\mathbf{1}}&{}\approx{}&0.010612418154&{}:{}&0.010612418154&{}:{}&0.978775163693&, \\ P^+_{\mathbf{1}}&{}\approx{}&0.074975567222&{}:{}&0.074975567222&{}:{}&0.850048865556&, \\ Q_{\mathbf{1}}&{}\approx{}&-0.075784598068&{}:{}&-0.075784598068&{}:{}&1.151569196137&, \\ I^\prime_{\mathbf{1}}&{}\approx{}&0.126120751079&{}:{}&0.126120751079&{}:{}&0.747758497842&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{1}}\)
\(P^-_{\mathbf{1}}\)
\(P^+_{\mathbf{1}}\)
\(Q_{\mathbf{1}}\)
\(I^\prime_{\mathbf{1}}\)
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1}}&{}\approx{}&0.076120467489&{}:{}&0.324423348821&{}:{}&0.599456183690&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.324423348821&{}:{}&0.076120467489&{}:{}&0.599456183690&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.042301124316&{}:{}&0.042301124316&{}:{}&0.915397751368&, \\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.074843504673&{}:{}&0.046202135415&{}:{}&0.878954359912&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.046202135415&{}:{}&0.074843504673&{}:{}&0.878954359912&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.200271908155&{}:{}&0.200271908155&{}:{}&0.599456183690&, \\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.043716483905&{}:{}&0.010257335682&{}:{}&0.946026180414&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.010257335682&{}:{}&0.043716483905&{}:{}&0.946026180414&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.101264812114&{}:{}&0.101264812114&{}:{}&0.797470375771&, \\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.070445884989&{}:{}&1.070445884989&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.070445884989&{}:{}&0.000000000000&{}:{}&1.070445884989&,\\C^*_{\mathbf{1}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1}}}{B^\prime_{\mathbf{1}}}{C^\prime_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1}}}{B^{\prime\prime}_{\mathbf{1}}}{C^{\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1}}}{B^{\prime\prime\prime}_{\mathbf{1}}}{C^{\prime\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^*_{\mathbf{1}}}{B^*_{\mathbf{1}}}{C^*_{\mathbf{1}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1}}}}&{}\approx{}&1.248302881333&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1}}}}&{}\approx{}&1.248302881333&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1}}}}&{}\approx{}&0.162764534548&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.259891532474&{}:{}&0.259891532474&{}:{}&0.480216935052&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&0.076120467489&{}:{}&0.324423348821&{}:{}&0.599456183690&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.324423348821&{}:{}&0.076120467489&{}:{}&0.599456183690&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.042301124316&{}:{}&0.042301124316&{}:{}&0.915397751368&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1}}&{}\approx{}&0.047564446284&{}:{}&0.047564446284&{}:{}&0.904871107431&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.074843504673&{}:{}&0.046202135415&{}:{}&0.878954359912&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.046202135415&{}:{}&0.074843504673&{}:{}&0.878954359912&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.200271908155&{}:{}&0.200271908155&{}:{}&0.599456183690&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1}}&{}\approx{}&0.010612418154&{}:{}&0.010612418154&{}:{}&0.978775163693&,\\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.043716483905&{}:{}&0.010257335682&{}:{}&0.946026180414&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.010257335682&{}:{}&0.043716483905&{}:{}&0.946026180414&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.101264812114&{}:{}&0.101264812114&{}:{}&0.797470375771&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1}}&{}\approx{}&0.074975567222&{}:{}&0.074975567222&{}:{}&0.850048865556&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&0.076120467489&{}:{}&0.324423348821&{}:{}&0.599456183690&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.324423348821&{}:{}&0.076120467489&{}:{}&0.599456183690&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.042301124316&{}:{}&0.042301124316&{}:{}&0.915397751368&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.074843504673&{}:{}&0.046202135415&{}:{}&0.878954359912&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.046202135415&{}:{}&0.074843504673&{}:{}&0.878954359912&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.200271908155&{}:{}&0.200271908155&{}:{}&0.599456183690&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1}}&{}\approx{}&-0.075784598068&{}:{}&-0.075784598068&{}:{}&1.151569196137&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.070445884989&{}:{}&1.070445884989&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.070445884989&{}:{}&0.000000000000&{}:{}&1.070445884989&,\\C^*_{\mathbf{1}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1}}&{}\approx{}&0.126120751079&{}:{}&0.126120751079&{}:{}&0.747758497842&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.074843504673&{}:{}&0.046202135415&{}:{}&0.878954359912&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.046202135415&{}:{}&0.074843504673&{}:{}&0.878954359912&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.200271908155&{}:{}&0.200271908155&{}:{}&0.599456183690&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.070445884989&{}:{}&1.070445884989&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.070445884989&{}:{}&0.000000000000&{}:{}&1.070445884989&,\\C^*_{\mathbf{1}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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