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 > 0 (000)

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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.259891532474&{}:{}&0.259891532474&{}:{}&0.480216935052&, \\ P&{}\approx{}&0.172650295333&{}:{}&0.172650295333&{}:{}&0.654699409335&, \\ P^-&{}\approx{}&0.145066082349&{}:{}&0.145066082349&{}:{}&0.709867835302&, \\ P^+&{}\approx{}&0.189548592918&{}:{}&0.189548592918&{}:{}&0.620902814163&, \\ Q&{}\approx{}&0.113841646514&{}:{}&0.113841646514&{}:{}&0.772316706973&, \\ I^\prime&{}\approx{}&0.215190892271&{}:{}&0.215190892271&{}:{}&0.569618215458&, \end{alignedat} \]
\(I\) Incenter
\(P\) First Ajima-Malfatti point
\(P^-\) First Malfatti-Rabinowitz point
\(P^+\) Gergonne point of the Malfatti triangle
\(Q\) Second Malfatti-Rabinowitz point
\(I^\prime\) Radical center of the Malfatti circles
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.382683432365&{}:{}&0.216772751325&{}:{}&0.400543816310&,\\B^\prime&{}\approx{}&0.216772751325&{}:{}&0.382683432365&{}:{}&0.400543816310&,\\C^\prime&{}\approx{}&0.093996764303&{}:{}&0.093996764303&{}:{}&0.812006471395&, \\ A^{\prime\prime}&{}\approx{}&0.131132148997&{}:{}&0.181314249869&{}:{}&0.687553601134&,\\B^{\prime\prime}&{}\approx{}&0.181314249869&{}:{}&0.131132148997&{}:{}&0.687553601134&,\\C^{\prime\prime}&{}\approx{}&0.299728091845&{}:{}&0.299728091845&{}:{}&0.400543816310&, \\ A^{\prime\prime\prime}&{}\approx{}&0.087688292400&{}:{}&0.154802005828&{}:{}&0.757509701772&,\\B^{\prime\prime\prime}&{}\approx{}&0.154802005828&{}:{}&0.087688292400&{}:{}&0.757509701772&,\\C^{\prime\prime\prime}&{}\approx{}&0.328227057688&{}:{}&0.328227057688&{}:{}&0.343545884624&, \\ A^*&{}\approx{}&0.000000000000&{}:{}&0.128466482391&{}:{}&0.871533517609&,\\B^*&{}\approx{}&0.128466482391&{}:{}&0.000000000000&{}:{}&0.871533517609&,\\C^*&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime}{B^\prime}{C^\prime}\)
\(\triangle{A^{\prime\prime}}{B^{\prime\prime}}{C^{\prime\prime}}\)
\(\triangle{A^{\prime\prime\prime}}{B^{\prime\prime\prime}}{C^{\prime\prime\prime}}\)
\(\triangle{A^*}{B^*}{C^*}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime}}&{}\approx{}&0.834089318960&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime}}&{}\approx{}&0.834089318960&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime}}&{}\approx{}&0.361676901928&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.259891532474&{}:{}&0.259891532474&{}:{}&0.480216935052&,\\ A^\prime&{}\approx{}&0.382683432365&{}:{}&0.216772751325&{}:{}&0.400543816310&,\\B^\prime&{}\approx{}&0.216772751325&{}:{}&0.382683432365&{}:{}&0.400543816310&,\\C^\prime&{}\approx{}&0.093996764303&{}:{}&0.093996764303&{}:{}&0.812006471395&. \end{alignedat} \]
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First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P&{}\approx{}&0.172650295333&{}:{}&0.172650295333&{}:{}&0.654699409335&,\\ A^{\prime\prime}&{}\approx{}&0.131132148997&{}:{}&0.181314249869&{}:{}&0.687553601134&,\\B^{\prime\prime}&{}\approx{}&0.181314249869&{}:{}&0.131132148997&{}:{}&0.687553601134&,\\C^{\prime\prime}&{}\approx{}&0.299728091845&{}:{}&0.299728091845&{}:{}&0.400543816310&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-&{}\approx{}&0.145066082349&{}:{}&0.145066082349&{}:{}&0.709867835302&,\\ A^{\prime\prime\prime}&{}\approx{}&0.087688292400&{}:{}&0.154802005828&{}:{}&0.757509701772&,\\B^{\prime\prime\prime}&{}\approx{}&0.154802005828&{}:{}&0.087688292400&{}:{}&0.757509701772&,\\C^{\prime\prime\prime}&{}\approx{}&0.328227057688&{}:{}&0.328227057688&{}:{}&0.343545884624&. \end{alignedat} \]
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Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+&{}\approx{}&0.189548592918&{}:{}&0.189548592918&{}:{}&0.620902814163&,\\ A^\prime&{}\approx{}&0.382683432365&{}:{}&0.216772751325&{}:{}&0.400543816310&,\\B^\prime&{}\approx{}&0.216772751325&{}:{}&0.382683432365&{}:{}&0.400543816310&,\\C^\prime&{}\approx{}&0.093996764303&{}:{}&0.093996764303&{}:{}&0.812006471395&,\\ A^{\prime\prime}&{}\approx{}&0.131132148997&{}:{}&0.181314249869&{}:{}&0.687553601134&,\\B^{\prime\prime}&{}\approx{}&0.181314249869&{}:{}&0.131132148997&{}:{}&0.687553601134&,\\C^{\prime\prime}&{}\approx{}&0.299728091845&{}:{}&0.299728091845&{}:{}&0.400543816310&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q&{}\approx{}&0.113841646514&{}:{}&0.113841646514&{}:{}&0.772316706973&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.128466482391&{}:{}&0.871533517609&,\\B^*&{}\approx{}&0.128466482391&{}:{}&0.000000000000&{}:{}&0.871533517609&,\\C^*&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.215190892271&{}:{}&0.215190892271&{}:{}&0.569618215458&,\\ A^{\prime\prime}&{}\approx{}&0.131132148997&{}:{}&0.181314249869&{}:{}&0.687553601134&,\\B^{\prime\prime}&{}\approx{}&0.181314249869&{}:{}&0.131132148997&{}:{}&0.687553601134&,\\C^{\prime\prime}&{}\approx{}&0.299728091845&{}:{}&0.299728091845&{}:{}&0.400543816310&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.128466482391&{}:{}&0.871533517609&,\\B^*&{}\approx{}&0.128466482391&{}:{}&0.000000000000&{}:{}&0.871533517609&,\\C^*&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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