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 > 3b (123)

3b (123)

Malfatti circles

3b (123)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.541196100146&{}:{}&-0.541196100146&{}:{}&1.000000000000&, \\ P_{\mathbf{3b}}&{}\approx{}&-0.000293086302&{}:{}&1.133780226419&{}:{}&-0.133487140117&, \\ P^-_{\mathbf{3b}}&{}\approx{}&0.097805468980&{}:{}&0.830334170838&{}:{}&0.071860360183&, \\ P^+_{\mathbf{3b}}&{}\approx{}&-0.154131852383&{}:{}&1.609646222900&{}:{}&-0.455514370517&, \\ Q_{\mathbf{3b}}&{}\approx{}&0.138010166320&{}:{}&1.707416774411&{}:{}&-0.845426940731&, \\ I^\prime_{\mathbf{3b}}&{}\approx{}&-0.031068625030&{}:{}&1.932362994164&{}:{}&-0.901294369134&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{3b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{3b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{3b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{3b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{3b}}\) Radical center of the Malfatti circles
3b (123)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3b}}&{}\approx{}&6.659529783962&{}:{}&6.675870559770&{}:{}&-12.335400343732&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.177900799442&{}:{}&1.506618614285&{}:{}&-0.328717814843&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-1.364388631412&{}:{}&1.364388631412&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.314760411920&{}:{}&1.490212596613&{}:{}&-0.175452184694&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.002772941545&{}:{}&2.265718474976&{}:{}&-1.262945533432&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.000424151453&{}:{}&1.640794968502&{}:{}&-0.640370817050&, \\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.121924481207&{}:{}&1.032562492694&{}:{}&0.089361988513&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.365339354347&{}:{}&0.366235803298&{}:{}&0.268424842355&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.130935353359&{}:{}&1.111595283972&{}:{}&-0.242530637331&, \\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&1.980785280403&{}:{}&-0.980785280403&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.195090322016&{}:{}&0.000000000000&{}:{}&1.195090322016&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.074784952616&{}:{}&0.925215047384&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3b}}}{B^\prime_{\mathbf{3b}}}{C^\prime_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^*_{\mathbf{3b}}}{B^*_{\mathbf{3b}}}{C^*_{\mathbf{3b}}}\)
3b (123)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3b}}}}&{}\approx{}&-12.335400343732&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3b}}}}&{}\approx{}&-0.328717814843&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3b}}}}&{}\approx{}&-2.521061461906&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.541196100146&{}:{}&-0.541196100146&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&6.659529783962&{}:{}&6.675870559770&{}:{}&-12.335400343732&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.177900799442&{}:{}&1.506618614285&{}:{}&-0.328717814843&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-1.364388631412&{}:{}&1.364388631412&{}:{}&1.000000000000&. \end{alignedat} \]
3b (123)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3b}}&{}\approx{}&-0.000293086302&{}:{}&1.133780226419&{}:{}&-0.133487140117&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.314760411920&{}:{}&1.490212596613&{}:{}&-0.175452184694&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.002772941545&{}:{}&2.265718474976&{}:{}&-1.262945533432&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.000424151453&{}:{}&1.640794968502&{}:{}&-0.640370817050&. \end{alignedat} \]
3b (123)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3b}}&{}\approx{}&0.097805468980&{}:{}&0.830334170838&{}:{}&0.071860360183&,\\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.121924481207&{}:{}&1.032562492694&{}:{}&0.089361988513&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.365339354347&{}:{}&0.366235803298&{}:{}&0.268424842355&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.130935353359&{}:{}&1.111595283972&{}:{}&-0.242530637331&. \end{alignedat} \]
3b (123)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3b}}&{}\approx{}&-0.154131852383&{}:{}&1.609646222900&{}:{}&-0.455514370517&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&6.659529783962&{}:{}&6.675870559770&{}:{}&-12.335400343732&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.177900799442&{}:{}&1.506618614285&{}:{}&-0.328717814843&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-1.364388631412&{}:{}&1.364388631412&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.314760411920&{}:{}&1.490212596613&{}:{}&-0.175452184694&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.002772941545&{}:{}&2.265718474976&{}:{}&-1.262945533432&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.000424151453&{}:{}&1.640794968502&{}:{}&-0.640370817050&, \end{alignedat} \]
3b (123)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3b}}&{}\approx{}&0.138010166320&{}:{}&1.707416774411&{}:{}&-0.845426940731&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&1.980785280403&{}:{}&-0.980785280403&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.195090322016&{}:{}&0.000000000000&{}:{}&1.195090322016&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.074784952616&{}:{}&0.925215047384&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3b}}&{}\approx{}&-0.031068625030&{}:{}&1.932362994164&{}:{}&-0.901294369134&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.314760411920&{}:{}&1.490212596613&{}:{}&-0.175452184694&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.002772941545&{}:{}&2.265718474976&{}:{}&-1.262945533432&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.000424151453&{}:{}&1.640794968502&{}:{}&-0.640370817050&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&1.980785280403&{}:{}&-0.980785280403&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.195090322016&{}:{}&0.000000000000&{}:{}&1.195090322016&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.074784952616&{}:{}&0.925215047384&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)