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 > 5a (213)

5a(213)

Malfatti circles

5a (213)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.541196100146&{}:{}&0.541196100146&{}:{}&1.000000000000&, \\ P_{\mathbf{5a}}&{}\approx{}&1.133780226419&{}:{}&-0.000293086302&{}:{}&-0.133487140117&, \\ P^-_{\mathbf{5a}}&{}\approx{}&0.830334170838&{}:{}&0.097805468980&{}:{}&0.071860360183&, \\ P^+_{\mathbf{5a}}&{}\approx{}&1.609646222900&{}:{}&-0.154131852383&{}:{}&-0.455514370517&, \\ Q_{\mathbf{5a}}&{}\approx{}&1.707416774411&{}:{}&0.138010166320&{}:{}&-0.845426940731&, \\ I^\prime_{\mathbf{5a}}&{}\approx{}&1.932362994164&{}:{}&-0.031068625030&{}:{}&-0.901294369134&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{5a}}\)
\(P^-_{\mathbf{5a}}\)
\(P^+_{\mathbf{5a}}\)
\(Q_{\mathbf{5a}}\)
\(I^\prime_{\mathbf{5a}}\)
5a (213)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5a}}&{}\approx{}&1.506618614285&{}:{}&-0.177900799442&{}:{}&-0.328717814843&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&6.675870559770&{}:{}&6.659529783962&{}:{}&-12.335400343732&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.364388631412&{}:{}&-1.364388631412&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&2.265718474976&{}:{}&-0.002772941545&{}:{}&-1.262945533432&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.490212596613&{}:{}&-0.314760411920&{}:{}&-0.175452184694&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.640794968502&{}:{}&-0.000424151453&{}:{}&-0.640370817050&, \\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.366235803298&{}:{}&0.365339354347&{}:{}&0.268424842355&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.032562492694&{}:{}&-0.121924481207&{}:{}&0.089361988513&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.111595283972&{}:{}&0.130935353359&{}:{}&-0.242530637331&, \\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.195090322016&{}:{}&1.195090322016&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.980785280403&{}:{}&0.000000000000&{}:{}&-0.980785280403&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.925215047384&{}:{}&0.074784952616&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5a}}}{B^\prime_{\mathbf{5a}}}{C^\prime_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^*_{\mathbf{5a}}}{B^*_{\mathbf{5a}}}{C^*_{\mathbf{5a}}}\)
5a (213)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5a}}}}&{}\approx{}&-0.328717814843&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5a}}}}&{}\approx{}&-12.335400343732&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5a}}}}&{}\approx{}&-2.521061461906&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.541196100146&{}:{}&0.541196100146&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.506618614285&{}:{}&-0.177900799442&{}:{}&-0.328717814843&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&6.675870559770&{}:{}&6.659529783962&{}:{}&-12.335400343732&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.364388631412&{}:{}&-1.364388631412&{}:{}&1.000000000000&. \end{alignedat} \]
5a (213)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5a}}&{}\approx{}&1.133780226419&{}:{}&-0.000293086302&{}:{}&-0.133487140117&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&2.265718474976&{}:{}&-0.002772941545&{}:{}&-1.262945533432&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.490212596613&{}:{}&-0.314760411920&{}:{}&-0.175452184694&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.640794968502&{}:{}&-0.000424151453&{}:{}&-0.640370817050&. \end{alignedat} \]
5a (213)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5a}}&{}\approx{}&0.830334170838&{}:{}&0.097805468980&{}:{}&0.071860360183&,\\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.366235803298&{}:{}&0.365339354347&{}:{}&0.268424842355&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.032562492694&{}:{}&-0.121924481207&{}:{}&0.089361988513&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.111595283972&{}:{}&0.130935353359&{}:{}&-0.242530637331&. \end{alignedat} \]
5a (213)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5a}}&{}\approx{}&1.609646222900&{}:{}&-0.154131852383&{}:{}&-0.455514370517&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.506618614285&{}:{}&-0.177900799442&{}:{}&-0.328717814843&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&6.675870559770&{}:{}&6.659529783962&{}:{}&-12.335400343732&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.364388631412&{}:{}&-1.364388631412&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&2.265718474976&{}:{}&-0.002772941545&{}:{}&-1.262945533432&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.490212596613&{}:{}&-0.314760411920&{}:{}&-0.175452184694&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.640794968502&{}:{}&-0.000424151453&{}:{}&-0.640370817050&, \end{alignedat} \]
5a (213)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5a}}&{}\approx{}&1.707416774411&{}:{}&0.138010166320&{}:{}&-0.845426940731&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.195090322016&{}:{}&1.195090322016&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.980785280403&{}:{}&0.000000000000&{}:{}&-0.980785280403&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.925215047384&{}:{}&0.074784952616&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5a}}&{}\approx{}&1.932362994164&{}:{}&-0.031068625030&{}:{}&-0.901294369134&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&2.265718474976&{}:{}&-0.002772941545&{}:{}&-1.262945533432&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.490212596613&{}:{}&-0.314760411920&{}:{}&-0.175452184694&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.640794968502&{}:{}&-0.000424151453&{}:{}&-0.640370817050&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.195090322016&{}:{}&1.195090322016&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.980785280403&{}:{}&0.000000000000&{}:{}&-0.980785280403&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.925215047384&{}:{}&0.074784952616&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)