Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).

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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.250000000000&{}:{}&0.333333333333&{}:{}&0.416666666667&, \\ P_{\mathbf{1}}&{}\approx{}&0.013125705415&{}:{}&0.018517744099&{}:{}&0.968356550486&, \\ P^-_{\mathbf{1}}&{}\approx{}&-0.010930768419&{}:{}&-0.013454289835&{}:{}&1.024385058254&, \\ P^+_{\mathbf{1}}&{}\approx{}&0.033120847037&{}:{}&0.045092102042&{}:{}&0.921787050921&, \\ Q_{\mathbf{1}}&{}\approx{}&-0.080500144965&{}:{}&-0.117104883017&{}:{}&1.197605027981&, \\ I^\prime_{\mathbf{1}}&{}\approx{}&0.074292983178&{}:{}&0.101894297245&{}:{}&0.823812719577&, \end{alignedat} \]
\(I\) Incenter
\(P_{\mathbf{1}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{1}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{1}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{1}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{1}}\) Radical center of the Malfatti circles
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1}}&{}\approx{}&-0.167552905630&{}:{}&0.518912402502&{}:{}&0.648640503128&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.468000242482&{}:{}&-0.248000646617&{}:{}&0.780000404136&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.023430099995&{}:{}&0.031240133326&{}:{}&0.945329766679&, \\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.044626029495&{}:{}&0.017926670906&{}:{}&0.937447299599&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.012585226673&{}:{}&0.058932356331&{}:{}&0.928482416996&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.121004919965&{}:{}&0.170713730949&{}:{}&0.708281349087&, \\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.024499659832&{}:{}&-0.012982752846&{}:{}&0.988483093015&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.010437024633&{}:{}&0.032323530929&{}:{}&0.978113493704&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.609609710931&{}:{}&-0.750346674921&{}:{}&2.359956385852&, \\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.108380256645&{}:{}&1.108380256645&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.072061402818&{}:{}&0.000000000000&{}:{}&1.072061402818&,\\C^*_{\mathbf{1}}&{}\approx{}&0.407379031733&{}:{}&0.592620968267&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1}}}{B^\prime_{\mathbf{1}}}{C^\prime_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1}}}{B^{\prime\prime}_{\mathbf{1}}}{C^{\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1}}}{B^{\prime\prime\prime}_{\mathbf{1}}}{C^{\prime\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^*_{\mathbf{1}}}{B^*_{\mathbf{1}}}{C^*_{\mathbf{1}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1}}}}&{}\approx{}&1.556737207507&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1}}}}&{}\approx{}&1.872000969926&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1}}}}&{}\approx{}&0.093720399979&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.250000000000&{}:{}&0.333333333333&{}:{}&0.416666666667&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.167552905630&{}:{}&0.518912402502&{}:{}&0.648640503128&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.468000242482&{}:{}&-0.248000646617&{}:{}&0.780000404136&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.023430099995&{}:{}&0.031240133326&{}:{}&0.945329766679&. \end{alignedat} \]
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First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1}}&{}\approx{}&0.013125705415&{}:{}&0.018517744099&{}:{}&0.968356550486&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.044626029495&{}:{}&0.017926670906&{}:{}&0.937447299599&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.012585226673&{}:{}&0.058932356331&{}:{}&0.928482416996&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.121004919965&{}:{}&0.170713730949&{}:{}&0.708281349087&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1}}&{}\approx{}&-0.010930768419&{}:{}&-0.013454289835&{}:{}&1.024385058254&,\\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.024499659832&{}:{}&-0.012982752846&{}:{}&0.988483093015&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.010437024633&{}:{}&0.032323530929&{}:{}&0.978113493704&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.609609710931&{}:{}&-0.750346674921&{}:{}&2.359956385852&. \end{alignedat} \]
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Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1}}&{}\approx{}&0.033120847037&{}:{}&0.045092102042&{}:{}&0.921787050921&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.167552905630&{}:{}&0.518912402502&{}:{}&0.648640503128&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.468000242482&{}:{}&-0.248000646617&{}:{}&0.780000404136&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.023430099995&{}:{}&0.031240133326&{}:{}&0.945329766679&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.044626029495&{}:{}&0.017926670906&{}:{}&0.937447299599&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.012585226673&{}:{}&0.058932356331&{}:{}&0.928482416996&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.121004919965&{}:{}&0.170713730949&{}:{}&0.708281349087&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1}}&{}\approx{}&-0.080500144965&{}:{}&-0.117104883017&{}:{}&1.197605027981&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.108380256645&{}:{}&1.108380256645&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.072061402818&{}:{}&0.000000000000&{}:{}&1.072061402818&,\\C^*_{\mathbf{1}}&{}\approx{}&0.407379031733&{}:{}&0.592620968267&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1}}&{}\approx{}&0.074292983178&{}:{}&0.101894297245&{}:{}&0.823812719577&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.044626029495&{}:{}&0.017926670906&{}:{}&0.937447299599&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.012585226673&{}:{}&0.058932356331&{}:{}&0.928482416996&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.121004919965&{}:{}&0.170713730949&{}:{}&0.708281349087&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.108380256645&{}:{}&1.108380256645&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.072061402818&{}:{}&0.000000000000&{}:{}&1.072061402818&,\\C^*_{\mathbf{1}}&{}\approx{}&0.407379031733&{}:{}&0.592620968267&{}:{}&0.000000000000&. \end{alignedat} \]
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