Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


[Top] > Pythagorean Triangle U > 0b (101)

0b(101)

Malfatti circles

0b (101)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.833333333333&{}:{}&-2.000000000000&{}:{}&2.166666666667&, \\ P_{\mathbf{b}}&{}\approx{}&0.797818964807&{}:{}&-0.816182936917&{}:{}&1.018363972109&, \\ P^-_{\mathbf{b}}&{}\approx{}&0.789860062090&{}:{}&-0.550885108697&{}:{}&0.761025046607&, \\ P^+_{\mathbf{b}}&{}\approx{}&0.803314657142&{}:{}&-0.999373420905&{}:{}&1.196058763763&, \\ Q_{\mathbf{b}}&{}\approx{}&0.880334064292&{}:{}&-0.325131054938&{}:{}&0.444796990646&, \\ I^\prime_{\mathbf{b}}&{}\approx{}&0.796928933401&{}:{}&-1.248724961858&{}:{}&1.451796028457&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{b}}\)
\(P^-_{\mathbf{b}}\)
\(P^+_{\mathbf{b}}\)
\(Q_{\mathbf{b}}\)
\(I^\prime_{\mathbf{b}}\)
0b (101)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{b}}&{}\approx{}&0.954847509134&{}:{}&-0.541829890387&{}:{}&0.586982381253&,\\B^\prime_{\mathbf{b}}&{}\approx{}&1.537998084118&{}:{}&-4.536793102824&{}:{}&3.998795018706&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.373827207244&{}:{}&-0.897185297387&{}:{}&1.523358090142&, \\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.601462415139&{}:{}&-1.608853056736&{}:{}&2.007390641597&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.736076943569&{}:{}&-0.675631244854&{}:{}&0.939554301285&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&1.029491148212&{}:{}&-1.053187685355&{}:{}&1.023696537143&, \\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.470537625590&{}:{}&-1.387993832008&{}:{}&1.917456206418&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.716310948722&{}:{}&-0.406471901656&{}:{}&0.690160952934&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&1.090164293995&{}:{}&-0.760331233873&{}:{}&0.670166939877&, \\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-2.716989200105&{}:{}&3.716989200105&,\\B^*_{\mathbf{b}}&{}\approx{}&0.664337358189&{}:{}&0.000000000000&{}:{}&0.335662641811&,\\C^*_{\mathbf{b}}&{}\approx{}&1.585607515558&{}:{}&-0.585607515558&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{b}}}{B^\prime_{\mathbf{b}}}{C^\prime_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{b}}}{B^{\prime\prime}_{\mathbf{b}}}{C^{\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{b}}}{B^{\prime\prime\prime}_{\mathbf{b}}}{C^{\prime\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^*_{\mathbf{b}}}{B^*_{\mathbf{b}}}{C^*_{\mathbf{b}}}\)
0b (101)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{b}}}}&{}\approx{}&0.270914945194&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{b}}}}&{}\approx{}&1.845597700941&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{b}}}}&{}\approx{}&0.448592648693&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.833333333333&{}:{}&-2.000000000000&{}:{}&2.166666666667&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.954847509134&{}:{}&-0.541829890387&{}:{}&0.586982381253&,\\B^\prime_{\mathbf{b}}&{}\approx{}&1.537998084118&{}:{}&-4.536793102824&{}:{}&3.998795018706&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.373827207244&{}:{}&-0.897185297387&{}:{}&1.523358090142&. \end{alignedat} \]
0b (101)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{b}}&{}\approx{}&0.797818964807&{}:{}&-0.816182936917&{}:{}&1.018363972109&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.601462415139&{}:{}&-1.608853056736&{}:{}&2.007390641597&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.736076943569&{}:{}&-0.675631244854&{}:{}&0.939554301285&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&1.029491148212&{}:{}&-1.053187685355&{}:{}&1.023696537143&. \end{alignedat} \]
0b (101)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{b}}&{}\approx{}&0.789860062090&{}:{}&-0.550885108697&{}:{}&0.761025046607&,\\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.470537625590&{}:{}&-1.387993832008&{}:{}&1.917456206418&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.716310948722&{}:{}&-0.406471901656&{}:{}&0.690160952934&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&1.090164293995&{}:{}&-0.760331233873&{}:{}&0.670166939877&. \end{alignedat} \]
0b (101)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{b}}&{}\approx{}&0.803314657142&{}:{}&-0.999373420905&{}:{}&1.196058763763&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.954847509134&{}:{}&-0.541829890387&{}:{}&0.586982381253&,\\B^\prime_{\mathbf{b}}&{}\approx{}&1.537998084118&{}:{}&-4.536793102824&{}:{}&3.998795018706&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.373827207244&{}:{}&-0.897185297387&{}:{}&1.523358090142&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.601462415139&{}:{}&-1.608853056736&{}:{}&2.007390641597&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.736076943569&{}:{}&-0.675631244854&{}:{}&0.939554301285&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&1.029491148212&{}:{}&-1.053187685355&{}:{}&1.023696537143&, \end{alignedat} \]
0b (101)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{b}}&{}\approx{}&0.880334064292&{}:{}&-0.325131054938&{}:{}&0.444796990646&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-2.716989200105&{}:{}&3.716989200105&,\\B^*_{\mathbf{b}}&{}\approx{}&0.664337358189&{}:{}&0.000000000000&{}:{}&0.335662641811&,\\C^*_{\mathbf{b}}&{}\approx{}&1.585607515558&{}:{}&-0.585607515558&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{b}}&{}\approx{}&0.796928933401&{}:{}&-1.248724961858&{}:{}&1.451796028457&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.601462415139&{}:{}&-1.608853056736&{}:{}&2.007390641597&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.736076943569&{}:{}&-0.675631244854&{}:{}&0.939554301285&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&1.029491148212&{}:{}&-1.053187685355&{}:{}&1.023696537143&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-2.716989200105&{}:{}&3.716989200105&,\\B^*_{\mathbf{b}}&{}\approx{}&0.664337358189&{}:{}&0.000000000000&{}:{}&0.335662641811&,\\C^*_{\mathbf{b}}&{}\approx{}&1.585607515558&{}:{}&-0.585607515558&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)