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 > 6b (321)

6b(321)

Malfatti circles

6b (321)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.541196100146&{}:{}&-0.541196100146&{}:{}&1.000000000000&, \\ P_{\mathbf{6b}}&{}\approx{}&-0.000570297495&{}:{}&1.000754750339&{}:{}&-0.000184452844&, \\ P^-_{\mathbf{6b}}&{}\approx{}&0.005574535548&{}:{}&0.983265606349&{}:{}&0.011159858103&, \\ P^+_{\mathbf{6b}}&{}\approx{}&-0.006857758045&{}:{}&1.018649834244&{}:{}&-0.011792076199&, \\ Q_{\mathbf{6b}}&{}\approx{}&-0.133409203006&{}:{}&1.111631062982&{}:{}&0.021778140024&, \\ I^\prime_{\mathbf{6b}}&{}\approx{}&-0.024926963029&{}:{}&1.044197063812&{}:{}&-0.019270100783&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{6b}}\)
\(P^-_{\mathbf{6b}}\)
\(P^+_{\mathbf{6b}}\)
\(Q_{\mathbf{6b}}\)
\(I^\prime_{\mathbf{6b}}\)
6b (321)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6b}}&{}\approx{}&15.038620480927&{}:{}&16.559681942832&{}:{}&-30.598302423759&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.006355829153&{}:{}&1.018099870087&{}:{}&-0.011744040934&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.550039664287&{}:{}&0.550039664287&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.012566450214&{}:{}&1.012753114521&{}:{}&-0.000186664307&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.048898757974&{}:{}&1.064714214022&{}:{}&-0.015815456048&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.000583579907&{}:{}&1.024062650191&{}:{}&-0.023479070284&, \\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.006211114771&{}:{}&0.994918993176&{}:{}&0.011292121596&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.243719427540&{}:{}&0.268370108047&{}:{}&0.487910464413&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.005702861783&{}:{}&1.005900455804&{}:{}&-0.011603317587&, \\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.980785280403&{}:{}&0.019214719597&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.195090322016&{}:{}&0.000000000000&{}:{}&-0.195090322016&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.136379290286&{}:{}&1.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6b}}}{B^\prime_{\mathbf{6b}}}{C^\prime_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^*_{\mathbf{6b}}}{B^*_{\mathbf{6b}}}{C^*_{\mathbf{6b}}}\)
6b (321)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6b}}}}&{}\approx{}&-30.598302423759&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6b}}}}&{}\approx{}&-0.011744040934&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6b}}}}&{}\approx{}&-1.016340775808&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.541196100146&{}:{}&-0.541196100146&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&15.038620480927&{}:{}&16.559681942832&{}:{}&-30.598302423759&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.006355829153&{}:{}&1.018099870087&{}:{}&-0.011744040934&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.550039664287&{}:{}&0.550039664287&{}:{}&1.000000000000&. \end{alignedat} \]
6b (321)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6b}}&{}\approx{}&-0.000570297495&{}:{}&1.000754750339&{}:{}&-0.000184452844&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.012566450214&{}:{}&1.012753114521&{}:{}&-0.000186664307&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.048898757974&{}:{}&1.064714214022&{}:{}&-0.015815456048&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.000583579907&{}:{}&1.024062650191&{}:{}&-0.023479070284&. \end{alignedat} \]
6b (321)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6b}}&{}\approx{}&0.005574535548&{}:{}&0.983265606349&{}:{}&0.011159858103&,\\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.006211114771&{}:{}&0.994918993176&{}:{}&0.011292121596&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.243719427540&{}:{}&0.268370108047&{}:{}&0.487910464413&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.005702861783&{}:{}&1.005900455804&{}:{}&-0.011603317587&. \end{alignedat} \]
6b (321)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6b}}&{}\approx{}&-0.006857758045&{}:{}&1.018649834244&{}:{}&-0.011792076199&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&15.038620480927&{}:{}&16.559681942832&{}:{}&-30.598302423759&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.006355829153&{}:{}&1.018099870087&{}:{}&-0.011744040934&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.550039664287&{}:{}&0.550039664287&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.012566450214&{}:{}&1.012753114521&{}:{}&-0.000186664307&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.048898757974&{}:{}&1.064714214022&{}:{}&-0.015815456048&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.000583579907&{}:{}&1.024062650191&{}:{}&-0.023479070284&, \end{alignedat} \]
6b (321)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6b}}&{}\approx{}&-0.133409203006&{}:{}&1.111631062982&{}:{}&0.021778140024&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.980785280403&{}:{}&0.019214719597&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.195090322016&{}:{}&0.000000000000&{}:{}&-0.195090322016&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.136379290286&{}:{}&1.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6b}}&{}\approx{}&-0.024926963029&{}:{}&1.044197063812&{}:{}&-0.019270100783&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.012566450214&{}:{}&1.012753114521&{}:{}&-0.000186664307&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.048898757974&{}:{}&1.064714214022&{}:{}&-0.015815456048&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.000583579907&{}:{}&1.024062650191&{}:{}&-0.023479070284&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.980785280403&{}:{}&0.019214719597&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.195090322016&{}:{}&0.000000000000&{}:{}&-0.195090322016&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.136379290286&{}:{}&1.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)