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,
\[ \begin{alignedat}{4} I&{}\approx{}&0.250000000000&{}:{}&0.333333333333&{}:{}&0.416666666667&, \\ P_{\mathbf{7}}&{}\approx{}&0.731961479201&{}:{}&0.230589900975&{}:{}&0.037448619824&, \\ P^-_{\mathbf{7}}&{}\approx{}&-0.087155651310&{}:{}&0.405207394538&{}:{}&0.681948256772&, \\ P^+_{\mathbf{7}}&{}\approx{}&0.390534343609&{}:{}&0.303374547311&{}:{}&0.306091109080&, \\ Q_{\mathbf{7}}&{}\approx{}&0.480956034150&{}:{}&0.330618412088&{}:{}&0.188425553762&, \\ I^\prime_{\mathbf{7}}&{}\approx{}&0.515433274099&{}:{}&0.334054901097&{}:{}&0.150511824804&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{7}}\)
\(P^-_{\mathbf{7}}\)
\(P^+_{\mathbf{7}}\)
\(Q_{\mathbf{7}}\)
\(I^\prime_{\mathbf{7}}\)
7 (222)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7}}&{}\approx{}&0.063999515037&{}:{}&0.416000215539&{}:{}&0.520000269424&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.583776452815&{}:{}&-0.556737207507&{}:{}&0.972960754692&,\\C^\prime_{\mathbf{7}}&{}\approx{}&1.373016509412&{}:{}&1.830688679216&{}:{}&-2.203705188628&, \\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.819233023619&{}:{}&0.155511375973&{}:{}&0.025255600408&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.306379024312&{}:{}&0.677946009273&{}:{}&0.015674966415&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.245038499524&{}:{}&0.077194504009&{}:{}&0.677766996467&, \\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-0.641563499862&{}:{}&0.611847685239&{}:{}&1.029715814624&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-3.082287839717&{}:{}&-20.035033155111&{}:{}&24.117320994828&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.223422375219&{}:{}&-1.038743870108&{}:{}&1.815321494890&, \\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.636975735854&{}:{}&0.363024264146&,\\B^*_{\mathbf{7}}&{}\approx{}&0.718508012224&{}:{}&0.000000000000&{}:{}&0.281491987776&,\\C^*_{\mathbf{7}}&{}\approx{}&0.592620968267&{}:{}&0.407379031733&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7}}}{B^\prime_{\mathbf{7}}}{C^\prime_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7}}}{B^{\prime\prime}_{\mathbf{7}}}{C^{\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7}}}{B^{\prime\prime\prime}_{\mathbf{7}}}{C^{\prime\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^*_{\mathbf{7}}}{B^*_{\mathbf{7}}}{C^*_{\mathbf{7}}}\)
7 (222)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7}}}}&{}\approx{}&1.248000646617&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7}}}}&{}\approx{}&2.335105811260&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7}}}}&{}\approx{}&5.492066037648&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.250000000000&{}:{}&0.333333333333&{}:{}&0.416666666667&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&0.063999515037&{}:{}&0.416000215539&{}:{}&0.520000269424&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.583776452815&{}:{}&-0.556737207507&{}:{}&0.972960754692&,\\C^\prime_{\mathbf{7}}&{}\approx{}&1.373016509412&{}:{}&1.830688679216&{}:{}&-2.203705188628&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7}}&{}\approx{}&0.731961479201&{}:{}&0.230589900975&{}:{}&0.037448619824&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.819233023619&{}:{}&0.155511375973&{}:{}&0.025255600408&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.306379024312&{}:{}&0.677946009273&{}:{}&0.015674966415&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.245038499524&{}:{}&0.077194504009&{}:{}&0.677766996467&. \end{alignedat} \]
7 (222)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7}}&{}\approx{}&-0.087155651310&{}:{}&0.405207394538&{}:{}&0.681948256772&,\\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-0.641563499862&{}:{}&0.611847685239&{}:{}&1.029715814624&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-3.082287839717&{}:{}&-20.035033155111&{}:{}&24.117320994828&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.223422375219&{}:{}&-1.038743870108&{}:{}&1.815321494890&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7}}&{}\approx{}&0.390534343609&{}:{}&0.303374547311&{}:{}&0.306091109080&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&0.063999515037&{}:{}&0.416000215539&{}:{}&0.520000269424&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.583776452815&{}:{}&-0.556737207507&{}:{}&0.972960754692&,\\C^\prime_{\mathbf{7}}&{}\approx{}&1.373016509412&{}:{}&1.830688679216&{}:{}&-2.203705188628&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.819233023619&{}:{}&0.155511375973&{}:{}&0.025255600408&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.306379024312&{}:{}&0.677946009273&{}:{}&0.015674966415&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.245038499524&{}:{}&0.077194504009&{}:{}&0.677766996467&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7}}&{}\approx{}&0.480956034150&{}:{}&0.330618412088&{}:{}&0.188425553762&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.636975735854&{}:{}&0.363024264146&,\\B^*_{\mathbf{7}}&{}\approx{}&0.718508012224&{}:{}&0.000000000000&{}:{}&0.281491987776&,\\C^*_{\mathbf{7}}&{}\approx{}&0.592620968267&{}:{}&0.407379031733&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7}}&{}\approx{}&0.515433274099&{}:{}&0.334054901097&{}:{}&0.150511824804&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.819233023619&{}:{}&0.155511375973&{}:{}&0.025255600408&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.306379024312&{}:{}&0.677946009273&{}:{}&0.015674966415&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.245038499524&{}:{}&0.077194504009&{}:{}&0.677766996467&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.636975735854&{}:{}&0.363024264146&,\\B^*_{\mathbf{7}}&{}\approx{}&0.718508012224&{}:{}&0.000000000000&{}:{}&0.281491987776&,\\C^*_{\mathbf{7}}&{}\approx{}&0.592620968267&{}:{}&0.407379031733&{}:{}&0.000000000000&. \end{alignedat} \]
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