Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


[Top] > Pythagorean Triangle U > 0a (011)

0a(011)

Malfatti circles

0a (011)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.250000000000&{}:{}&0.600000000000&{}:{}&0.650000000000&, \\ P_{\mathbf{a}}&{}\approx{}&-0.156379250510&{}:{}&0.609090936413&{}:{}&0.547288314097&, \\ P^-_{\mathbf{a}}&{}\approx{}&-0.127875918644&{}:{}&0.611858720039&{}:{}&0.516017198604&, \\ P^+_{\mathbf{a}}&{}\approx{}&-0.174095171106&{}:{}&0.607370652090&{}:{}&0.566724519016&, \\ Q_{\mathbf{a}}&{}\approx{}&-0.116441789457&{}:{}&0.629348219008&{}:{}&0.487093570449&, \\ I^\prime_{\mathbf{a}}&{}\approx{}&-0.197085219125&{}:{}&0.602575098265&{}:{}&0.594510120861&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{a}}\)
\(P^-_{\mathbf{a}}\)
\(P^+_{\mathbf{a}}\)
\(Q_{\mathbf{a}}\)
\(I^\prime_{\mathbf{a}}\)
0a (011)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{a}}&{}\approx{}&-0.506052104307&{}:{}&0.722905010067&{}:{}&0.783147094240&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.103748070570&{}:{}&0.834003087088&{}:{}&0.269744983482&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.122256164685&{}:{}&0.293414795243&{}:{}&0.828841369441&, \\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.112244278848&{}:{}&0.585844055075&{}:{}&0.526400223773&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.233059286351&{}:{}&0.417410001066&{}:{}&0.815649285284&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.206816084122&{}:{}&0.805540389357&{}:{}&0.401275694765&, \\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.072367906342&{}:{}&0.581746310689&{}:{}&0.490621595653&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.224023618489&{}:{}&0.320021979557&{}:{}&0.904001638932&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.187534691880&{}:{}&0.897313096584&{}:{}&0.290221595296&, \\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.563708941166&{}:{}&0.436291058834&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.314154134496&{}:{}&0.000000000000&{}:{}&1.314154134496&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.227023454471&{}:{}&1.227023454471&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{a}}}{B^\prime_{\mathbf{a}}}{C^\prime_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{a}}}{B^{\prime\prime}_{\mathbf{a}}}{C^{\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{a}}}{B^{\prime\prime\prime}_{\mathbf{a}}}{C^{\prime\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^*_{\mathbf{a}}}{B^*_{\mathbf{a}}}{C^*_{\mathbf{a}}}\)
0a (011)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{a}}}}&{}\approx{}&1.204841683445&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{a}}}}&{}\approx{}&0.414992282281&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{a}}}}&{}\approx{}&0.489024658739&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.250000000000&{}:{}&0.600000000000&{}:{}&0.650000000000&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-0.506052104307&{}:{}&0.722905010067&{}:{}&0.783147094240&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.103748070570&{}:{}&0.834003087088&{}:{}&0.269744983482&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.122256164685&{}:{}&0.293414795243&{}:{}&0.828841369441&. \end{alignedat} \]
0a (011)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{a}}&{}\approx{}&-0.156379250510&{}:{}&0.609090936413&{}:{}&0.547288314097&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.112244278848&{}:{}&0.585844055075&{}:{}&0.526400223773&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.233059286351&{}:{}&0.417410001066&{}:{}&0.815649285284&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.206816084122&{}:{}&0.805540389357&{}:{}&0.401275694765&. \end{alignedat} \]
0a (011)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{a}}&{}\approx{}&-0.127875918644&{}:{}&0.611858720039&{}:{}&0.516017198604&,\\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.072367906342&{}:{}&0.581746310689&{}:{}&0.490621595653&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.224023618489&{}:{}&0.320021979557&{}:{}&0.904001638932&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.187534691880&{}:{}&0.897313096584&{}:{}&0.290221595296&. \end{alignedat} \]
0a (011)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{a}}&{}\approx{}&-0.174095171106&{}:{}&0.607370652090&{}:{}&0.566724519016&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-0.506052104307&{}:{}&0.722905010067&{}:{}&0.783147094240&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.103748070570&{}:{}&0.834003087088&{}:{}&0.269744983482&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.122256164685&{}:{}&0.293414795243&{}:{}&0.828841369441&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.112244278848&{}:{}&0.585844055075&{}:{}&0.526400223773&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.233059286351&{}:{}&0.417410001066&{}:{}&0.815649285284&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.206816084122&{}:{}&0.805540389357&{}:{}&0.401275694765&, \end{alignedat} \]
0a (011)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{a}}&{}\approx{}&-0.116441789457&{}:{}&0.629348219008&{}:{}&0.487093570449&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.563708941166&{}:{}&0.436291058834&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.314154134496&{}:{}&0.000000000000&{}:{}&1.314154134496&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.227023454471&{}:{}&1.227023454471&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{a}}&{}\approx{}&-0.197085219125&{}:{}&0.602575098265&{}:{}&0.594510120861&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.112244278848&{}:{}&0.585844055075&{}:{}&0.526400223773&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.233059286351&{}:{}&0.417410001066&{}:{}&0.815649285284&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.206816084122&{}:{}&0.805540389357&{}:{}&0.401275694765&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.563708941166&{}:{}&0.436291058834&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.314154134496&{}:{}&0.000000000000&{}:{}&1.314154134496&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.227023454471&{}:{}&1.227023454471&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)