Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 135° Top Angle

\(A=22.5\degree\), \(B=22.5\degree\), \(C=135\degree\).


[Top] > Isosceles Triangle with 135° Top Angle > 5b (303)

5b(303)

Malfatti circles

5b (303)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.541196100146&{}:{}&-0.541196100146&{}:{}&1.000000000000&, \\ P_{\mathbf{5b}}&{}\approx{}&0.004820719120&{}:{}&-0.000796033466&{}:{}&0.995975314347&, \\ P^-_{\mathbf{5b}}&{}\approx{}&-0.008242636546&{}:{}&0.012365342917&{}:{}&0.995877293628&, \\ P^+_{\mathbf{5b}}&{}\approx{}&0.017277317002&{}:{}&-0.013346099268&{}:{}&0.996068782266&, \\ Q_{\mathbf{5b}}&{}\approx{}&0.192061679716&{}:{}&0.015524308276&{}:{}&0.792414012009&, \\ I^\prime_{\mathbf{5b}}&{}\approx{}&0.049671834969&{}:{}&-0.020184590973&{}:{}&0.970512756004&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{5b}}\)
\(P^-_{\mathbf{5b}}\)
\(P^+_{\mathbf{5b}}\)
\(Q_{\mathbf{5b}}\)
\(I^\prime_{\mathbf{5b}}\)
5b (303)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5b}}&{}\approx{}&-2.811770718939&{}:{}&-4.496290132666&{}:{}&8.308060851606&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.264138361741&{}:{}&0.247797585933&{}:{}&0.488064052326&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.013235329055&{}:{}&-0.013235329055&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.025207568343&{}:{}&-0.000779726239&{}:{}&0.975572157896&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.004944016600&{}:{}&-0.026392967481&{}:{}&1.021448950881&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.093467678127&{}:{}&-0.015434087312&{}:{}&0.921966409185&, \\ A^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.012904309493&{}:{}&0.012105991418&{}:{}&0.974989699089&,\\B^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&-0.008458724011&{}:{}&-0.013526308191&{}:{}&1.021985032202&,\\C^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&-0.289442992798&{}:{}&0.434213232761&{}:{}&0.855229760037&, \\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.019214719597&{}:{}&0.980785280403&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.195090322016&{}:{}&0.000000000000&{}:{}&0.804909677984&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.925215047384&{}:{}&0.074784952616&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5b}}}{B^\prime_{\mathbf{5b}}}{C^\prime_{\mathbf{5b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5b}}}{B^{\prime\prime}_{\mathbf{5b}}}{C^{\prime\prime}_{\mathbf{5b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5b}}}{B^{\prime\prime\prime}_{\mathbf{5b}}}{C^{\prime\prime\prime}_{\mathbf{5b}}}\)
\(\triangle{A^*_{\mathbf{5b}}}{B^*_{\mathbf{5b}}}{C^*_{\mathbf{5b}}}\)
5b (303)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5b}}}}&{}\approx{}&8.308060851606&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5b}}}}&{}\approx{}&0.488064052326&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5b}}}}&{}\approx{}&0.024455699240&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.541196100146&{}:{}&-0.541196100146&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{5b}}&{}\approx{}&-2.811770718939&{}:{}&-4.496290132666&{}:{}&8.308060851606&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.264138361741&{}:{}&0.247797585933&{}:{}&0.488064052326&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.013235329055&{}:{}&-0.013235329055&{}:{}&1.000000000000&. \end{alignedat} \]
5b (303)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5b}}&{}\approx{}&0.004820719120&{}:{}&-0.000796033466&{}:{}&0.995975314347&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.025207568343&{}:{}&-0.000779726239&{}:{}&0.975572157896&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.004944016600&{}:{}&-0.026392967481&{}:{}&1.021448950881&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.093467678127&{}:{}&-0.015434087312&{}:{}&0.921966409185&. \end{alignedat} \]
5b (303)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5b}}&{}\approx{}&-0.008242636546&{}:{}&0.012365342917&{}:{}&0.995877293628&,\\ A^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.012904309493&{}:{}&0.012105991418&{}:{}&0.974989699089&,\\B^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&-0.008458724011&{}:{}&-0.013526308191&{}:{}&1.021985032202&,\\C^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&-0.289442992798&{}:{}&0.434213232761&{}:{}&0.855229760037&. \end{alignedat} \]
5b (303)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5b}}&{}\approx{}&0.017277317002&{}:{}&-0.013346099268&{}:{}&0.996068782266&,\\ A^\prime_{\mathbf{5b}}&{}\approx{}&-2.811770718939&{}:{}&-4.496290132666&{}:{}&8.308060851606&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.264138361741&{}:{}&0.247797585933&{}:{}&0.488064052326&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.013235329055&{}:{}&-0.013235329055&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.025207568343&{}:{}&-0.000779726239&{}:{}&0.975572157896&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.004944016600&{}:{}&-0.026392967481&{}:{}&1.021448950881&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.093467678127&{}:{}&-0.015434087312&{}:{}&0.921966409185&, \end{alignedat} \]
5b (303)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5b}}&{}\approx{}&0.192061679716&{}:{}&0.015524308276&{}:{}&0.792414012009&,\\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.019214719597&{}:{}&0.980785280403&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.195090322016&{}:{}&0.000000000000&{}:{}&0.804909677984&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.925215047384&{}:{}&0.074784952616&{}:{}&0.000000000000&. \end{alignedat} \]
5b (303)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5b}}&{}\approx{}&0.049671834969&{}:{}&-0.020184590973&{}:{}&0.970512756004&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.025207568343&{}:{}&-0.000779726239&{}:{}&0.975572157896&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.004944016600&{}:{}&-0.026392967481&{}:{}&1.021448950881&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.093467678127&{}:{}&-0.015434087312&{}:{}&0.921966409185&,\\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.019214719597&{}:{}&0.980785280403&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.195090322016&{}:{}&0.000000000000&{}:{}&0.804909677984&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.925215047384&{}:{}&0.074784952616&{}:{}&0.000000000000&. \end{alignedat} \]
5b (303)