Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.166666666667&{}:{}&0.400000000000&{}:{}&0.433333333333&, \\ P_{\mathbf{3}}&{}\approx{}&0.002204336719&{}:{}&0.735725412682&{}:{}&0.262070250599&, \\ P^-_{\mathbf{3}}&{}\approx{}&-0.057475331931&{}:{}&0.857552589716&{}:{}&0.199922742215&, \\ P^+_{\mathbf{3}}&{}\approx{}&0.036786112845&{}:{}&0.665131854479&{}:{}&0.298082032676&, \\ Q_{\mathbf{3}}&{}\approx{}&-0.017625498021&{}:{}&0.587894884100&{}:{}&0.429730613921&, \\ I^\prime_{\mathbf{3}}&{}\approx{}&0.021329234185&{}:{}&0.603670279334&{}:{}&0.375000486481&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{3}}\)
\(P^-_{\mathbf{3}}\)
\(P^+_{\mathbf{3}}\)
\(Q_{\mathbf{3}}\)
\(I^\prime_{\mathbf{3}}\)
3 (022)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3}}&{}\approx{}&-0.234667392808&{}:{}&0.592640348548&{}:{}&0.642027044260&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.056245467280&{}:{}&0.797516317790&{}:{}&0.146238214929&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.149128546814&{}:{}&0.357908512353&{}:{}&0.492962940833&, \\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.081683214280&{}:{}&0.677121600153&{}:{}&0.241195185567&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.004618527974&{}:{}&0.446291683119&{}:{}&0.549089788907&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.002275592466&{}:{}&0.759507924455&{}:{}&0.238216483079&, \\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.054098400978&{}:{}&0.767072612821&{}:{}&0.178828986201&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.199845809199&{}:{}&0.504700242341&{}:{}&0.695145566859&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.060037482729&{}:{}&0.895780799600&{}:{}&0.164256683129&, \\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.577712415072&{}:{}&0.422287584928&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.042769422997&{}:{}&0.000000000000&{}:{}&1.042769422997&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.030907319333&{}:{}&1.030907319333&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3}}}{B^\prime_{\mathbf{3}}}{C^\prime_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3}}}{B^{\prime\prime}_{\mathbf{3}}}{C^{\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3}}}{B^{\prime\prime\prime}_{\mathbf{3}}}{C^{\prime\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^*_{\mathbf{3}}}{B^*_{\mathbf{3}}}{C^*_{\mathbf{3}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3}}}}&{}\approx{}&1.481600871370&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3}}}}&{}\approx{}&0.337472803683&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3}}}}&{}\approx{}&0.894771280883&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.166666666667&{}:{}&0.400000000000&{}:{}&0.433333333333&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.234667392808&{}:{}&0.592640348548&{}:{}&0.642027044260&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.056245467280&{}:{}&0.797516317790&{}:{}&0.146238214929&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.149128546814&{}:{}&0.357908512353&{}:{}&0.492962940833&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3}}&{}\approx{}&0.002204336719&{}:{}&0.735725412682&{}:{}&0.262070250599&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.081683214280&{}:{}&0.677121600153&{}:{}&0.241195185567&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.004618527974&{}:{}&0.446291683119&{}:{}&0.549089788907&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.002275592466&{}:{}&0.759507924455&{}:{}&0.238216483079&. \end{alignedat} \]
3 (022)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3}}&{}\approx{}&-0.057475331931&{}:{}&0.857552589716&{}:{}&0.199922742215&,\\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.054098400978&{}:{}&0.767072612821&{}:{}&0.178828986201&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.199845809199&{}:{}&0.504700242341&{}:{}&0.695145566859&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.060037482729&{}:{}&0.895780799600&{}:{}&0.164256683129&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3}}&{}\approx{}&0.036786112845&{}:{}&0.665131854479&{}:{}&0.298082032676&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.234667392808&{}:{}&0.592640348548&{}:{}&0.642027044260&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.056245467280&{}:{}&0.797516317790&{}:{}&0.146238214929&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.149128546814&{}:{}&0.357908512353&{}:{}&0.492962940833&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.081683214280&{}:{}&0.677121600153&{}:{}&0.241195185567&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.004618527974&{}:{}&0.446291683119&{}:{}&0.549089788907&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.002275592466&{}:{}&0.759507924455&{}:{}&0.238216483079&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3}}&{}\approx{}&-0.017625498021&{}:{}&0.587894884100&{}:{}&0.429730613921&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.577712415072&{}:{}&0.422287584928&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.042769422997&{}:{}&0.000000000000&{}:{}&1.042769422997&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.030907319333&{}:{}&1.030907319333&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3}}&{}\approx{}&0.021329234185&{}:{}&0.603670279334&{}:{}&0.375000486481&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.081683214280&{}:{}&0.677121600153&{}:{}&0.241195185567&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.004618527974&{}:{}&0.446291683119&{}:{}&0.549089788907&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.002275592466&{}:{}&0.759507924455&{}:{}&0.238216483079&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.577712415072&{}:{}&0.422287584928&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.042769422997&{}:{}&0.000000000000&{}:{}&1.042769422997&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.030907319333&{}:{}&1.030907319333&{}:{}&0.000000000000&. \end{alignedat} \]
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