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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[Top] > Isosceles Triangle with 108° Top Angle > 0 (000)

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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.276393202250&{}:{}&0.276393202250&{}:{}&0.447213595500&, \\ P&{}\approx{}&0.225067351166&{}:{}&0.225067351166&{}:{}&0.549865297667&, \\ P^-&{}\approx{}&0.207367247767&{}:{}&0.207367247767&{}:{}&0.585265504466&, \\ P^+&{}\approx{}&0.235542551970&{}:{}&0.235542551970&{}:{}&0.528914896060&, \\ Q&{}\approx{}&0.191680415255&{}:{}&0.191680415255&{}:{}&0.616639169491&, \\ I^\prime&{}\approx{}&0.250738053924&{}:{}&0.250738053924&{}:{}&0.498523892152&, \end{alignedat} \]
\(I\)
\(P\)
\(P^-\)
\(P^+\)
\(Q\)
\(I^\prime\)
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.453848561685&{}:{}&0.208611286432&{}:{}&0.337540151884&,\\B^\prime&{}\approx{}&0.208611286432&{}:{}&0.453848561685&{}:{}&0.337540151884&,\\C^\prime&{}\approx{}&0.122846377163&{}:{}&0.122846377163&{}:{}&0.754307245674&, \\ A^{\prime\prime}&{}\approx{}&0.154632965764&{}:{}&0.245523942558&{}:{}&0.599843091678&,\\B^{\prime\prime}&{}\approx{}&0.245523942558&{}:{}&0.154632965764&{}:{}&0.599843091678&,\\C^{\prime\prime}&{}\approx{}&0.331229924058&{}:{}&0.331229924058&{}:{}&0.337540151884&, \\ A^{\prime\prime\prime}&{}\approx{}&0.107344296000&{}:{}&0.233535084212&{}:{}&0.659120619788&,\\B^{\prime\prime\prime}&{}\approx{}&0.233535084212&{}:{}&0.107344296000&{}:{}&0.659120619788&,\\C^{\prime\prime\prime}&{}\approx{}&0.364467536191&{}:{}&0.364467536191&{}:{}&0.271064927618&, \\ A^*&{}\approx{}&0.000000000000&{}:{}&0.237134443940&{}:{}&0.762865556060&,\\B^*&{}\approx{}&0.237134443940&{}:{}&0.000000000000&{}:{}&0.762865556060&,\\C^*&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime}{B^\prime}{C^\prime}\)
\(\triangle{A^{\prime\prime}}{B^{\prime\prime}}{C^{\prime\prime}}\)
\(\triangle{A^{\prime\prime\prime}}{B^{\prime\prime\prime}}{C^{\prime\prime\prime}}\)
\(\triangle{A^*}{B^*}{C^*}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime}}&{}\approx{}&0.754762724747&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime}}&{}\approx{}&0.754762724747&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime}}&{}\approx{}&0.444462367970&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.276393202250&{}:{}&0.276393202250&{}:{}&0.447213595500&,\\ A^\prime&{}\approx{}&0.453848561685&{}:{}&0.208611286432&{}:{}&0.337540151884&,\\B^\prime&{}\approx{}&0.208611286432&{}:{}&0.453848561685&{}:{}&0.337540151884&,\\C^\prime&{}\approx{}&0.122846377163&{}:{}&0.122846377163&{}:{}&0.754307245674&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P&{}\approx{}&0.225067351166&{}:{}&0.225067351166&{}:{}&0.549865297667&,\\ A^{\prime\prime}&{}\approx{}&0.154632965764&{}:{}&0.245523942558&{}:{}&0.599843091678&,\\B^{\prime\prime}&{}\approx{}&0.245523942558&{}:{}&0.154632965764&{}:{}&0.599843091678&,\\C^{\prime\prime}&{}\approx{}&0.331229924058&{}:{}&0.331229924058&{}:{}&0.337540151884&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-&{}\approx{}&0.207367247767&{}:{}&0.207367247767&{}:{}&0.585265504466&,\\ A^{\prime\prime\prime}&{}\approx{}&0.107344296000&{}:{}&0.233535084212&{}:{}&0.659120619788&,\\B^{\prime\prime\prime}&{}\approx{}&0.233535084212&{}:{}&0.107344296000&{}:{}&0.659120619788&,\\C^{\prime\prime\prime}&{}\approx{}&0.364467536191&{}:{}&0.364467536191&{}:{}&0.271064927618&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+&{}\approx{}&0.235542551970&{}:{}&0.235542551970&{}:{}&0.528914896060&,\\ A^\prime&{}\approx{}&0.453848561685&{}:{}&0.208611286432&{}:{}&0.337540151884&,\\B^\prime&{}\approx{}&0.208611286432&{}:{}&0.453848561685&{}:{}&0.337540151884&,\\C^\prime&{}\approx{}&0.122846377163&{}:{}&0.122846377163&{}:{}&0.754307245674&,\\ A^{\prime\prime}&{}\approx{}&0.154632965764&{}:{}&0.245523942558&{}:{}&0.599843091678&,\\B^{\prime\prime}&{}\approx{}&0.245523942558&{}:{}&0.154632965764&{}:{}&0.599843091678&,\\C^{\prime\prime}&{}\approx{}&0.331229924058&{}:{}&0.331229924058&{}:{}&0.337540151884&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q&{}\approx{}&0.191680415255&{}:{}&0.191680415255&{}:{}&0.616639169491&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.237134443940&{}:{}&0.762865556060&,\\B^*&{}\approx{}&0.237134443940&{}:{}&0.000000000000&{}:{}&0.762865556060&,\\C^*&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.250738053924&{}:{}&0.250738053924&{}:{}&0.498523892152&,\\ A^{\prime\prime}&{}\approx{}&0.154632965764&{}:{}&0.245523942558&{}:{}&0.599843091678&,\\B^{\prime\prime}&{}\approx{}&0.245523942558&{}:{}&0.154632965764&{}:{}&0.599843091678&,\\C^{\prime\prime}&{}\approx{}&0.331229924058&{}:{}&0.331229924058&{}:{}&0.337540151884&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.237134443940&{}:{}&0.762865556060&,\\B^*&{}\approx{}&0.237134443940&{}:{}&0.000000000000&{}:{}&0.762865556060&,\\C^*&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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