Derousseau's Generalization of the Malfatti circles

Equilateral Triangle

\(a:b:c=1:1:1\), \(A=60\degree\), \(B=60\degree\), \(C=60\degree\).


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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&, \\ P&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&, \\ P^-&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&, \\ P^+&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&, \\ Q&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&, \\ I^\prime&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&, \end{alignedat} \]
\(I\)
\(P\)
\(P^-\)
\(P^+\)
\(Q\)
\(I^\prime\)
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.577350269190&{}:{}&0.211324865405&{}:{}&0.211324865405&,\\B^\prime&{}\approx{}&0.211324865405&{}:{}&0.577350269190&{}:{}&0.211324865405&,\\C^\prime&{}\approx{}&0.211324865405&{}:{}&0.211324865405&{}:{}&0.577350269190&, \\ A^{\prime\prime}&{}\approx{}&0.211324865405&{}:{}&0.394337567297&{}:{}&0.394337567297&,\\B^{\prime\prime}&{}\approx{}&0.394337567297&{}:{}&0.211324865405&{}:{}&0.394337567297&,\\C^{\prime\prime}&{}\approx{}&0.394337567297&{}:{}&0.394337567297&{}:{}&0.211324865405&, \\ A^{\prime\prime\prime}&{}\approx{}&0.154700538379&{}:{}&0.422649730810&{}:{}&0.422649730810&,\\B^{\prime\prime\prime}&{}\approx{}&0.422649730810&{}:{}&0.154700538379&{}:{}&0.422649730810&,\\C^{\prime\prime\prime}&{}\approx{}&0.422649730810&{}:{}&0.422649730810&{}:{}&0.154700538379&, \\ A^*&{}\approx{}&0.000000000000&{}:{}&0.500000000000&{}:{}&0.500000000000&,\\B^*&{}\approx{}&0.500000000000&{}:{}&0.000000000000&{}:{}&0.500000000000&,\\C^*&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime}{B^\prime}{C^\prime}\)
\(\triangle{A^{\prime\prime}}{B^{\prime\prime}}{C^{\prime\prime}}\)
\(\triangle{A^{\prime\prime\prime}}{B^{\prime\prime\prime}}{C^{\prime\prime\prime}}\)
\(\triangle{A^*}{B^*}{C^*}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime}}&{}\approx{}&0.633974596216&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime}}&{}\approx{}&0.633974596216&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime}}&{}\approx{}&0.633974596216&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&,\\ A^\prime&{}\approx{}&0.577350269190&{}:{}&0.211324865405&{}:{}&0.211324865405&,\\B^\prime&{}\approx{}&0.211324865405&{}:{}&0.577350269190&{}:{}&0.211324865405&,\\C^\prime&{}\approx{}&0.211324865405&{}:{}&0.211324865405&{}:{}&0.577350269190&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&,\\ A^{\prime\prime}&{}\approx{}&0.211324865405&{}:{}&0.394337567297&{}:{}&0.394337567297&,\\B^{\prime\prime}&{}\approx{}&0.394337567297&{}:{}&0.211324865405&{}:{}&0.394337567297&,\\C^{\prime\prime}&{}\approx{}&0.394337567297&{}:{}&0.394337567297&{}:{}&0.211324865405&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&,\\ A^{\prime\prime\prime}&{}\approx{}&0.154700538379&{}:{}&0.422649730810&{}:{}&0.422649730810&,\\B^{\prime\prime\prime}&{}\approx{}&0.422649730810&{}:{}&0.154700538379&{}:{}&0.422649730810&,\\C^{\prime\prime\prime}&{}\approx{}&0.422649730810&{}:{}&0.422649730810&{}:{}&0.154700538379&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&,\\ A^\prime&{}\approx{}&0.577350269190&{}:{}&0.211324865405&{}:{}&0.211324865405&,\\B^\prime&{}\approx{}&0.211324865405&{}:{}&0.577350269190&{}:{}&0.211324865405&,\\C^\prime&{}\approx{}&0.211324865405&{}:{}&0.211324865405&{}:{}&0.577350269190&,\\ A^{\prime\prime}&{}\approx{}&0.211324865405&{}:{}&0.394337567297&{}:{}&0.394337567297&,\\B^{\prime\prime}&{}\approx{}&0.394337567297&{}:{}&0.211324865405&{}:{}&0.394337567297&,\\C^{\prime\prime}&{}\approx{}&0.394337567297&{}:{}&0.394337567297&{}:{}&0.211324865405&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.500000000000&{}:{}&0.500000000000&,\\B^*&{}\approx{}&0.500000000000&{}:{}&0.000000000000&{}:{}&0.500000000000&,\\C^*&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&,\\ A^{\prime\prime}&{}\approx{}&0.211324865405&{}:{}&0.394337567297&{}:{}&0.394337567297&,\\B^{\prime\prime}&{}\approx{}&0.394337567297&{}:{}&0.211324865405&{}:{}&0.394337567297&,\\C^{\prime\prime}&{}\approx{}&0.394337567297&{}:{}&0.394337567297&{}:{}&0.211324865405&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.500000000000&{}:{}&0.500000000000&,\\B^*&{}\approx{}&0.500000000000&{}:{}&0.000000000000&{}:{}&0.500000000000&,\\C^*&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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