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,
\[ \begin{alignedat}{4} I&{}\approx{}&0.250000000000&{}:{}&0.333333333333&{}:{}&0.416666666667&, \\ P_{\mathbf{5}}&{}\approx{}&0.950443520472&{}:{}&0.000929879531&{}:{}&0.048626599997&, \\ P^-_{\mathbf{5}}&{}\approx{}&1.099850274582&{}:{}&-0.069972798038&{}:{}&-0.029877476544&, \\ P^+_{\mathbf{5}}&{}\approx{}&0.845714717145&{}:{}&0.050630126476&{}:{}&0.103655156379&, \\ Q_{\mathbf{5}}&{}\approx{}&0.738844717962&{}:{}&-0.028304076492&{}:{}&0.289459358529&, \\ I^\prime_{\mathbf{5}}&{}\approx{}&0.752939500973&{}:{}&0.027194409752&{}:{}&0.219866089275&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{5}}\)
\(P^-_{\mathbf{5}}\)
\(P^+_{\mathbf{5}}\)
\(Q_{\mathbf{5}}\)
\(I^\prime_{\mathbf{5}}\)
5 (202)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5}}&{}\approx{}&0.873016509412&{}:{}&0.056437106928&{}:{}&0.070546383660&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.738284950003&{}:{}&-0.968759866674&{}:{}&1.230474916671&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.563999515037&{}:{}&0.751999353383&{}:{}&-0.315998868420&, \\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.639479875463&{}:{}&0.006764812343&{}:{}&0.353755312194&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.851443972781&{}:{}&0.104994442978&{}:{}&0.043561584241&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.865710845562&{}:{}&0.000846980150&{}:{}&0.133442174288&, \\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&-1.146186633268&{}:{}&1.503998706765&{}:{}&0.642187926503&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.963873023464&{}:{}&0.062310625634&{}:{}&-0.026183649098&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.983103089917&{}:{}&-0.062545307803&{}:{}&0.079442217886&, \\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.108380256645&{}:{}&1.108380256645&,\\B^*_{\mathbf{5}}&{}\approx{}&0.718508012224&{}:{}&0.000000000000&{}:{}&0.281491987776&,\\C^*_{\mathbf{5}}&{}\approx{}&1.039834563767&{}:{}&-0.039834563767&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5}}}{B^\prime_{\mathbf{5}}}{C^\prime_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5}}}{B^{\prime\prime}_{\mathbf{5}}}{C^{\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5}}}{B^{\prime\prime\prime}_{\mathbf{5}}}{C^{\prime\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^*_{\mathbf{5}}}{B^*_{\mathbf{5}}}{C^*_{\mathbf{5}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5}}}}&{}\approx{}&0.169311320784&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5}}}}&{}\approx{}&2.953139800010&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5}}}}&{}\approx{}&2.255998060148&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.250000000000&{}:{}&0.333333333333&{}:{}&0.416666666667&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.873016509412&{}:{}&0.056437106928&{}:{}&0.070546383660&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.738284950003&{}:{}&-0.968759866674&{}:{}&1.230474916671&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.563999515037&{}:{}&0.751999353383&{}:{}&-0.315998868420&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5}}&{}\approx{}&0.950443520472&{}:{}&0.000929879531&{}:{}&0.048626599997&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.639479875463&{}:{}&0.006764812343&{}:{}&0.353755312194&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.851443972781&{}:{}&0.104994442978&{}:{}&0.043561584241&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.865710845562&{}:{}&0.000846980150&{}:{}&0.133442174288&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5}}&{}\approx{}&1.099850274582&{}:{}&-0.069972798038&{}:{}&-0.029877476544&,\\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&-1.146186633268&{}:{}&1.503998706765&{}:{}&0.642187926503&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.963873023464&{}:{}&0.062310625634&{}:{}&-0.026183649098&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.983103089917&{}:{}&-0.062545307803&{}:{}&0.079442217886&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5}}&{}\approx{}&0.845714717145&{}:{}&0.050630126476&{}:{}&0.103655156379&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.873016509412&{}:{}&0.056437106928&{}:{}&0.070546383660&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.738284950003&{}:{}&-0.968759866674&{}:{}&1.230474916671&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.563999515037&{}:{}&0.751999353383&{}:{}&-0.315998868420&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.639479875463&{}:{}&0.006764812343&{}:{}&0.353755312194&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.851443972781&{}:{}&0.104994442978&{}:{}&0.043561584241&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.865710845562&{}:{}&0.000846980150&{}:{}&0.133442174288&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5}}&{}\approx{}&0.738844717962&{}:{}&-0.028304076492&{}:{}&0.289459358529&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.108380256645&{}:{}&1.108380256645&,\\B^*_{\mathbf{5}}&{}\approx{}&0.718508012224&{}:{}&0.000000000000&{}:{}&0.281491987776&,\\C^*_{\mathbf{5}}&{}\approx{}&1.039834563767&{}:{}&-0.039834563767&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5}}&{}\approx{}&0.752939500973&{}:{}&0.027194409752&{}:{}&0.219866089275&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.639479875463&{}:{}&0.006764812343&{}:{}&0.353755312194&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.851443972781&{}:{}&0.104994442978&{}:{}&0.043561584241&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.865710845562&{}:{}&0.000846980150&{}:{}&0.133442174288&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.108380256645&{}:{}&1.108380256645&,\\B^*_{\mathbf{5}}&{}\approx{}&0.718508012224&{}:{}&0.000000000000&{}:{}&0.281491987776&,\\C^*_{\mathbf{5}}&{}\approx{}&1.039834563767&{}:{}&-0.039834563767&{}:{}&0.000000000000&. \end{alignedat} \]
5 (202)