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 > 3a (033)

3a(033)

Malfatti circles

3a (033)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.541196100146&{}:{}&0.541196100146&{}:{}&1.000000000000&, \\ P_{\mathbf{3a}}&{}\approx{}&-0.000796033466&{}:{}&0.004820719120&{}:{}&0.995975314347&, \\ P^-_{\mathbf{3a}}&{}\approx{}&0.012365342917&{}:{}&-0.008242636546&{}:{}&0.995877293628&, \\ P^+_{\mathbf{3a}}&{}\approx{}&-0.013346099268&{}:{}&0.017277317002&{}:{}&0.996068782266&, \\ Q_{\mathbf{3a}}&{}\approx{}&0.015524308276&{}:{}&0.192061679716&{}:{}&0.792414012009&, \\ I^\prime_{\mathbf{3a}}&{}\approx{}&-0.020184590973&{}:{}&0.049671834969&{}:{}&0.970512756004&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{3a}}\)
\(P^-_{\mathbf{3a}}\)
\(P^+_{\mathbf{3a}}\)
\(Q_{\mathbf{3a}}\)
\(I^\prime_{\mathbf{3a}}\)
3a (033)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3a}}&{}\approx{}&0.247797585933&{}:{}&0.264138361741&{}:{}&0.488064052326&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-4.496290132666&{}:{}&-2.811770718939&{}:{}&8.308060851606&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.013235329055&{}:{}&0.013235329055&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.026392967481&{}:{}&0.004944016600&{}:{}&1.021448950881&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.000779726239&{}:{}&0.025207568343&{}:{}&0.975572157896&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.015434087312&{}:{}&0.093467678127&{}:{}&0.921966409185&, \\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.013526308191&{}:{}&-0.008458724011&{}:{}&1.021985032202&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.012105991418&{}:{}&0.012904309493&{}:{}&0.974989699089&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.434213232761&{}:{}&-0.289442992798&{}:{}&0.855229760037&, \\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.195090322016&{}:{}&0.804909677984&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.019214719597&{}:{}&0.000000000000&{}:{}&0.980785280403&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.074784952616&{}:{}&0.925215047384&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3a}}}{B^\prime_{\mathbf{3a}}}{C^\prime_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^*_{\mathbf{3a}}}{B^*_{\mathbf{3a}}}{C^*_{\mathbf{3a}}}\)
3a (033)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3a}}}}&{}\approx{}&0.488064052326&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3a}}}}&{}\approx{}&8.308060851606&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3a}}}}&{}\approx{}&0.024455699240&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.541196100146&{}:{}&0.541196100146&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.247797585933&{}:{}&0.264138361741&{}:{}&0.488064052326&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-4.496290132666&{}:{}&-2.811770718939&{}:{}&8.308060851606&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.013235329055&{}:{}&0.013235329055&{}:{}&1.000000000000&. \end{alignedat} \]
3a (033)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3a}}&{}\approx{}&-0.000796033466&{}:{}&0.004820719120&{}:{}&0.995975314347&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.026392967481&{}:{}&0.004944016600&{}:{}&1.021448950881&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.000779726239&{}:{}&0.025207568343&{}:{}&0.975572157896&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.015434087312&{}:{}&0.093467678127&{}:{}&0.921966409185&. \end{alignedat} \]
3a (033)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3a}}&{}\approx{}&0.012365342917&{}:{}&-0.008242636546&{}:{}&0.995877293628&,\\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.013526308191&{}:{}&-0.008458724011&{}:{}&1.021985032202&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.012105991418&{}:{}&0.012904309493&{}:{}&0.974989699089&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.434213232761&{}:{}&-0.289442992798&{}:{}&0.855229760037&. \end{alignedat} \]
3a (033)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3a}}&{}\approx{}&-0.013346099268&{}:{}&0.017277317002&{}:{}&0.996068782266&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.247797585933&{}:{}&0.264138361741&{}:{}&0.488064052326&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-4.496290132666&{}:{}&-2.811770718939&{}:{}&8.308060851606&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.013235329055&{}:{}&0.013235329055&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.026392967481&{}:{}&0.004944016600&{}:{}&1.021448950881&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.000779726239&{}:{}&0.025207568343&{}:{}&0.975572157896&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.015434087312&{}:{}&0.093467678127&{}:{}&0.921966409185&, \end{alignedat} \]
3a (033)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3a}}&{}\approx{}&0.015524308276&{}:{}&0.192061679716&{}:{}&0.792414012009&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.195090322016&{}:{}&0.804909677984&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.019214719597&{}:{}&0.000000000000&{}:{}&0.980785280403&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.074784952616&{}:{}&0.925215047384&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3a}}&{}\approx{}&-0.020184590973&{}:{}&0.049671834969&{}:{}&0.970512756004&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.026392967481&{}:{}&0.004944016600&{}:{}&1.021448950881&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.000779726239&{}:{}&0.025207568343&{}:{}&0.975572157896&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.015434087312&{}:{}&0.093467678127&{}:{}&0.921966409185&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.195090322016&{}:{}&0.804909677984&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.019214719597&{}:{}&0.000000000000&{}:{}&0.980785280403&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.074784952616&{}:{}&0.925215047384&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)