Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 90° Top Angle

\(A=45\degree\), \(B=45\degree\), \(C=90\degree\).

[Top] > Isosceles Triangle with 90° Top Angle > 0 (000)

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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.292893218813&{}:{}&0.292893218813&{}:{}&0.414213562373&, \\ P&{}\approx{}&0.263422843010&{}:{}&0.263422843010&{}:{}&0.473154313981&, \\ P^-&{}\approx{}&0.252883322329&{}:{}&0.252883322329&{}:{}&0.494233355341&, \\ P^+&{}\approx{}&0.269567397243&{}:{}&0.269567397243&{}:{}&0.460865205513&, \\ Q&{}\approx{}&0.244954416660&{}:{}&0.244954416660&{}:{}&0.510091166680&, \\ I^\prime&{}\approx{}&0.278257463541&{}:{}&0.278257463541&{}:{}&0.443485072918&, \end{alignedat} \]
\(I\)
\(P\)
\(P^-\)
\(P^+\)
\(Q\)
\(I^\prime\)
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.500000000000&{}:{}&0.207106781187&{}:{}&0.292893218813&,\\B^\prime&{}\approx{}&0.207106781187&{}:{}&0.500000000000&{}:{}&0.292893218813&,\\C^\prime&{}\approx{}&0.148846697643&{}:{}&0.148846697643&{}:{}&0.702306604714&, \\ A^{\prime\prime}&{}\approx{}&0.173208929102&{}:{}&0.295686137432&{}:{}&0.531104933466&,\\B^{\prime\prime}&{}\approx{}&0.295686137432&{}:{}&0.173208929102&{}:{}&0.531104933466&,\\C^{\prime\prime}&{}\approx{}&0.353553390593&{}:{}&0.353553390593&{}:{}&0.292893218813&, \\ A^{\prime\prime\prime}&{}\approx{}&0.122962885091&{}:{}&0.296858664855&{}:{}&0.580178450054&,\\B^{\prime\prime\prime}&{}\approx{}&0.296858664855&{}:{}&0.122962885091&{}:{}&0.580178450054&,\\C^{\prime\prime\prime}&{}\approx{}&0.386729540170&{}:{}&0.386729540170&{}:{}&0.226540919661&, \\ A^*&{}\approx{}&0.000000000000&{}:{}&0.324423348821&{}:{}&0.675576651179&,\\B^*&{}\approx{}&0.324423348821&{}:{}&0.000000000000&{}:{}&0.675576651179&,\\C^*&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime}{B^\prime}{C^\prime}\)
\(\triangle{A^{\prime\prime}}{B^{\prime\prime}}{C^{\prime\prime}}\)
\(\triangle{A^{\prime\prime\prime}}{B^{\prime\prime\prime}}{C^{\prime\prime\prime}}\)
\(\triangle{A^*}{B^*}{C^*}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime}}&{}\approx{}&0.707106781187&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime}}&{}\approx{}&0.707106781187&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime}}&{}\approx{}&0.508194413807&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.292893218813&{}:{}&0.292893218813&{}:{}&0.414213562373&,\\ A^\prime&{}\approx{}&0.500000000000&{}:{}&0.207106781187&{}:{}&0.292893218813&,\\B^\prime&{}\approx{}&0.207106781187&{}:{}&0.500000000000&{}:{}&0.292893218813&,\\C^\prime&{}\approx{}&0.148846697643&{}:{}&0.148846697643&{}:{}&0.702306604714&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P&{}\approx{}&0.263422843010&{}:{}&0.263422843010&{}:{}&0.473154313981&,\\ A^{\prime\prime}&{}\approx{}&0.173208929102&{}:{}&0.295686137432&{}:{}&0.531104933466&,\\B^{\prime\prime}&{}\approx{}&0.295686137432&{}:{}&0.173208929102&{}:{}&0.531104933466&,\\C^{\prime\prime}&{}\approx{}&0.353553390593&{}:{}&0.353553390593&{}:{}&0.292893218813&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-&{}\approx{}&0.252883322329&{}:{}&0.252883322329&{}:{}&0.494233355341&,\\ A^{\prime\prime\prime}&{}\approx{}&0.122962885091&{}:{}&0.296858664855&{}:{}&0.580178450054&,\\B^{\prime\prime\prime}&{}\approx{}&0.296858664855&{}:{}&0.122962885091&{}:{}&0.580178450054&,\\C^{\prime\prime\prime}&{}\approx{}&0.386729540170&{}:{}&0.386729540170&{}:{}&0.226540919661&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+&{}\approx{}&0.269567397243&{}:{}&0.269567397243&{}:{}&0.460865205513&,\\ A^\prime&{}\approx{}&0.500000000000&{}:{}&0.207106781187&{}:{}&0.292893218813&,\\B^\prime&{}\approx{}&0.207106781187&{}:{}&0.500000000000&{}:{}&0.292893218813&,\\C^\prime&{}\approx{}&0.148846697643&{}:{}&0.148846697643&{}:{}&0.702306604714&,\\ A^{\prime\prime}&{}\approx{}&0.173208929102&{}:{}&0.295686137432&{}:{}&0.531104933466&,\\B^{\prime\prime}&{}\approx{}&0.295686137432&{}:{}&0.173208929102&{}:{}&0.531104933466&,\\C^{\prime\prime}&{}\approx{}&0.353553390593&{}:{}&0.353553390593&{}:{}&0.292893218813&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q&{}\approx{}&0.244954416660&{}:{}&0.244954416660&{}:{}&0.510091166680&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.324423348821&{}:{}&0.675576651179&,\\B^*&{}\approx{}&0.324423348821&{}:{}&0.000000000000&{}:{}&0.675576651179&,\\C^*&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.278257463541&{}:{}&0.278257463541&{}:{}&0.443485072918&,\\ A^{\prime\prime}&{}\approx{}&0.173208929102&{}:{}&0.295686137432&{}:{}&0.531104933466&,\\B^{\prime\prime}&{}\approx{}&0.295686137432&{}:{}&0.173208929102&{}:{}&0.531104933466&,\\C^{\prime\prime}&{}\approx{}&0.353553390593&{}:{}&0.353553390593&{}:{}&0.292893218813&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.324423348821&{}:{}&0.675576651179&,\\B^*&{}\approx{}&0.324423348821&{}:{}&0.000000000000&{}:{}&0.675576651179&,\\C^*&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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