Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


[Top] > Pythagorean Triangle U > 2b (121)

2b(121)

Malfatti circles

2b (121)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.833333333333&{}:{}&-2.000000000000&{}:{}&2.166666666667&, \\ P_{\mathbf{2b}}&{}\approx{}&-0.029070964164&{}:{}&1.598449083631&{}:{}&-0.569378119468&, \\ P^-_{\mathbf{2b}}&{}\approx{}&0.162225773979&{}:{}&0.800248548144&{}:{}&0.037525677878&, \\ P^+_{\mathbf{2b}}&{}\approx{}&-0.372904662315&{}:{}&3.033121962265&{}:{}&-1.660217299950&, \\ Q_{\mathbf{2b}}&{}\approx{}&3.682517731657&{}:{}&-9.970889437661&{}:{}&7.288371706004&, \\ I^\prime_{\mathbf{2b}}&{}\approx{}&-0.298378035089&{}:{}&3.427615958124&{}:{}&-2.129237923035&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{2b}}\)
\(P^-_{\mathbf{2b}}\)
\(P^+_{\mathbf{2b}}\)
\(Q_{\mathbf{2b}}\)
\(I^\prime_{\mathbf{2b}}\)
2b (121)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2b}}&{}\approx{}&1.295152490866&{}:{}&3.541829890387&{}:{}&-3.836982381253&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-1.371331417451&{}:{}&5.936793102824&{}:{}&-3.565461685373&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.623827207244&{}:{}&1.497185297387&{}:{}&0.126641909858&, \\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.857549708345&{}:{}&2.885319606231&{}:{}&-1.027769897886&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.053662585204&{}:{}&2.104687301042&{}:{}&-1.051024715838&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.086992606621&{}:{}&4.783234967840&{}:{}&-3.696242361219&, \\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.283107567176&{}:{}&1.225634467917&{}:{}&0.057473099259&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.252143257710&{}:{}&0.689531567221&{}:{}&0.058325175069&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.336659632898&{}:{}&1.660718736846&{}:{}&-0.997378369745&, \\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&3.716989200105&{}:{}&-2.716989200105&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.335662641811&{}:{}&0.000000000000&{}:{}&0.664337358189&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.585607515558&{}:{}&1.585607515558&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2b}}}{B^\prime_{\mathbf{2b}}}{C^\prime_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^*_{\mathbf{2b}}}{B^*_{\mathbf{2b}}}{C^*_{\mathbf{2b}}}\)
2b (121)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2b}}}}&{}\approx{}&-1.770914945194&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2b}}}}&{}\approx{}&-1.645597700941&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2b}}}}&{}\approx{}&-0.748592648693&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.833333333333&{}:{}&-2.000000000000&{}:{}&2.166666666667&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&1.295152490866&{}:{}&3.541829890387&{}:{}&-3.836982381253&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-1.371331417451&{}:{}&5.936793102824&{}:{}&-3.565461685373&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.623827207244&{}:{}&1.497185297387&{}:{}&0.126641909858&. \end{alignedat} \]
2b (121)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2b}}&{}\approx{}&-0.029070964164&{}:{}&1.598449083631&{}:{}&-0.569378119468&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.857549708345&{}:{}&2.885319606231&{}:{}&-1.027769897886&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.053662585204&{}:{}&2.104687301042&{}:{}&-1.051024715838&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.086992606621&{}:{}&4.783234967840&{}:{}&-3.696242361219&. \end{alignedat} \]
2b (121)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2b}}&{}\approx{}&0.162225773979&{}:{}&0.800248548144&{}:{}&0.037525677878&,\\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.283107567176&{}:{}&1.225634467917&{}:{}&0.057473099259&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.252143257710&{}:{}&0.689531567221&{}:{}&0.058325175069&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.336659632898&{}:{}&1.660718736846&{}:{}&-0.997378369745&. \end{alignedat} \]
2b (121)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2b}}&{}\approx{}&-0.372904662315&{}:{}&3.033121962265&{}:{}&-1.660217299950&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&1.295152490866&{}:{}&3.541829890387&{}:{}&-3.836982381253&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-1.371331417451&{}:{}&5.936793102824&{}:{}&-3.565461685373&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.623827207244&{}:{}&1.497185297387&{}:{}&0.126641909858&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.857549708345&{}:{}&2.885319606231&{}:{}&-1.027769897886&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.053662585204&{}:{}&2.104687301042&{}:{}&-1.051024715838&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.086992606621&{}:{}&4.783234967840&{}:{}&-3.696242361219&, \end{alignedat} \]
2b (121)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2b}}&{}\approx{}&3.682517731657&{}:{}&-9.970889437661&{}:{}&7.288371706004&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&3.716989200105&{}:{}&-2.716989200105&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.335662641811&{}:{}&0.000000000000&{}:{}&0.664337358189&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.585607515558&{}:{}&1.585607515558&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2b}}&{}\approx{}&-0.298378035089&{}:{}&3.427615958124&{}:{}&-2.129237923035&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.857549708345&{}:{}&2.885319606231&{}:{}&-1.027769897886&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.053662585204&{}:{}&2.104687301042&{}:{}&-1.051024715838&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.086992606621&{}:{}&4.783234967840&{}:{}&-3.696242361219&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&3.716989200105&{}:{}&-2.716989200105&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.335662641811&{}:{}&0.000000000000&{}:{}&0.664337358189&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.585607515558&{}:{}&1.585607515558&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)