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,
\[ \begin{alignedat}{4} I&{}\approx{}&0.250000000000&{}:{}&0.333333333333&{}:{}&0.416666666667&, \\ P_{\mathbf{4}}&{}\approx{}&0.997521724384&{}:{}&0.000975939141&{}:{}&0.001502336475&, \\ P^-_{\mathbf{4}}&{}\approx{}&1.021959809881&{}:{}&-0.009889535600&{}:{}&-0.012070274281&, \\ P^+_{\mathbf{4}}&{}\approx{}&0.974583442003&{}:{}&0.011174582887&{}:{}&0.014241975110&, \\ Q_{\mathbf{4}}&{}\approx{}&1.117975688967&{}:{}&-0.042828037674&{}:{}&-0.075147651292&, \\ I^\prime_{\mathbf{4}}&{}\approx{}&0.920624914364&{}:{}&0.033250813799&{}:{}&0.046124271837&, \end{alignedat} \]
\(I\) Incenter
\(P_{\mathbf{4}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{4}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{4}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{4}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{4}}\) Radical center of the Malfatti circles
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4}}&{}\approx{}&0.976569900005&{}:{}&0.010413377775&{}:{}&0.013016722219&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.686508254706&{}:{}&-0.830688679216&{}:{}&1.144180424510&,\\C^\prime_{\mathbf{4}}&{}\approx{}&1.167552905630&{}:{}&1.556737207507&{}:{}&-1.724290113137&, \\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.864625962367&{}:{}&0.053309979403&{}:{}&0.082064058231&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.977838940903&{}:{}&0.020688366250&{}:{}&0.001472692846&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.973307144066&{}:{}&0.000952248473&{}:{}&0.025740607461&, \\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-0.592817058853&{}:{}&0.717320463752&{}:{}&0.875496595101&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&1.001149009633&{}:{}&0.010675470181&{}:{}&-0.011824479814&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.996361295834&{}:{}&-0.009641818015&{}:{}&0.013280522181&, \\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.363024264146&{}:{}&0.636975735854&,\\B^*_{\mathbf{4}}&{}\approx{}&1.072061402818&{}:{}&0.000000000000&{}:{}&-0.072061402818&,\\C^*_{\mathbf{4}}&{}\approx{}&1.039834563767&{}:{}&-0.039834563767&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4}}}{B^\prime_{\mathbf{4}}}{C^\prime_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4}}}{B^{\prime\prime}_{\mathbf{4}}}{C^{\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4}}}{B^{\prime\prime\prime}_{\mathbf{4}}}{C^{\prime\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^*_{\mathbf{4}}}{B^*_{\mathbf{4}}}{C^*_{\mathbf{4}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4}}}}&{}\approx{}&0.031240133326&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4}}}}&{}\approx{}&2.746033018824&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4}}}}&{}\approx{}&4.670211622521&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.250000000000&{}:{}&0.333333333333&{}:{}&0.416666666667&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.976569900005&{}:{}&0.010413377775&{}:{}&0.013016722219&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.686508254706&{}:{}&-0.830688679216&{}:{}&1.144180424510&,\\C^\prime_{\mathbf{4}}&{}\approx{}&1.167552905630&{}:{}&1.556737207507&{}:{}&-1.724290113137&. \end{alignedat} \]
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First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4}}&{}\approx{}&0.997521724384&{}:{}&0.000975939141&{}:{}&0.001502336475&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.864625962367&{}:{}&0.053309979403&{}:{}&0.082064058231&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.977838940903&{}:{}&0.020688366250&{}:{}&0.001472692846&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.973307144066&{}:{}&0.000952248473&{}:{}&0.025740607461&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4}}&{}\approx{}&1.021959809881&{}:{}&-0.009889535600&{}:{}&-0.012070274281&,\\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-0.592817058853&{}:{}&0.717320463752&{}:{}&0.875496595101&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&1.001149009633&{}:{}&0.010675470181&{}:{}&-0.011824479814&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.996361295834&{}:{}&-0.009641818015&{}:{}&0.013280522181&. \end{alignedat} \]
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Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4}}&{}\approx{}&0.974583442003&{}:{}&0.011174582887&{}:{}&0.014241975110&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.976569900005&{}:{}&0.010413377775&{}:{}&0.013016722219&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.686508254706&{}:{}&-0.830688679216&{}:{}&1.144180424510&,\\C^\prime_{\mathbf{4}}&{}\approx{}&1.167552905630&{}:{}&1.556737207507&{}:{}&-1.724290113137&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.864625962367&{}:{}&0.053309979403&{}:{}&0.082064058231&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.977838940903&{}:{}&0.020688366250&{}:{}&0.001472692846&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.973307144066&{}:{}&0.000952248473&{}:{}&0.025740607461&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4}}&{}\approx{}&1.117975688967&{}:{}&-0.042828037674&{}:{}&-0.075147651292&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.363024264146&{}:{}&0.636975735854&,\\B^*_{\mathbf{4}}&{}\approx{}&1.072061402818&{}:{}&0.000000000000&{}:{}&-0.072061402818&,\\C^*_{\mathbf{4}}&{}\approx{}&1.039834563767&{}:{}&-0.039834563767&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4}}&{}\approx{}&0.920624914364&{}:{}&0.033250813799&{}:{}&0.046124271837&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.864625962367&{}:{}&0.053309979403&{}:{}&0.082064058231&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.977838940903&{}:{}&0.020688366250&{}:{}&0.001472692846&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.973307144066&{}:{}&0.000952248473&{}:{}&0.025740607461&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.363024264146&{}:{}&0.636975735854&,\\B^*_{\mathbf{4}}&{}\approx{}&1.072061402818&{}:{}&0.000000000000&{}:{}&-0.072061402818&,\\C^*_{\mathbf{4}}&{}\approx{}&1.039834563767&{}:{}&-0.039834563767&{}:{}&0.000000000000&. \end{alignedat} \]
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