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 > 6a (231)

6a(231)

Malfatti circles

6a (231)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.541196100146&{}:{}&0.541196100146&{}:{}&1.000000000000&, \\ P_{\mathbf{6a}}&{}\approx{}&1.000754750339&{}:{}&-0.000570297495&{}:{}&-0.000184452844&, \\ P^-_{\mathbf{6a}}&{}\approx{}&0.983265606349&{}:{}&0.005574535548&{}:{}&0.011159858103&, \\ P^+_{\mathbf{6a}}&{}\approx{}&1.018649834244&{}:{}&-0.006857758045&{}:{}&-0.011792076199&, \\ Q_{\mathbf{6a}}&{}\approx{}&1.111631062982&{}:{}&-0.133409203006&{}:{}&0.021778140024&, \\ I^\prime_{\mathbf{6a}}&{}\approx{}&1.044197063812&{}:{}&-0.024926963029&{}:{}&-0.019270100783&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{6a}}\)
\(P^-_{\mathbf{6a}}\)
\(P^+_{\mathbf{6a}}\)
\(Q_{\mathbf{6a}}\)
\(I^\prime_{\mathbf{6a}}\)
6a (231)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6a}}&{}\approx{}&1.018099870087&{}:{}&-0.006355829153&{}:{}&-0.011744040934&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&16.559681942832&{}:{}&15.038620480927&{}:{}&-30.598302423759&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.550039664287&{}:{}&-0.550039664287&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.064714214022&{}:{}&-0.048898757974&{}:{}&-0.015815456048&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.012753114521&{}:{}&-0.012566450214&{}:{}&-0.000186664307&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.024062650191&{}:{}&-0.000583579907&{}:{}&-0.023479070284&, \\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.268370108047&{}:{}&0.243719427540&{}:{}&0.487910464413&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.994918993176&{}:{}&-0.006211114771&{}:{}&0.011292121596&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.005900455804&{}:{}&0.005702861783&{}:{}&-0.011603317587&, \\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.195090322016&{}:{}&-0.195090322016&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.980785280403&{}:{}&0.000000000000&{}:{}&0.019214719597&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.136379290286&{}:{}&-0.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6a}}}{B^\prime_{\mathbf{6a}}}{C^\prime_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^*_{\mathbf{6a}}}{B^*_{\mathbf{6a}}}{C^*_{\mathbf{6a}}}\)
6a (231)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6a}}}}&{}\approx{}&-0.011744040934&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6a}}}}&{}\approx{}&-30.598302423759&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6a}}}}&{}\approx{}&-1.016340775808&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.541196100146&{}:{}&0.541196100146&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.018099870087&{}:{}&-0.006355829153&{}:{}&-0.011744040934&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&16.559681942832&{}:{}&15.038620480927&{}:{}&-30.598302423759&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.550039664287&{}:{}&-0.550039664287&{}:{}&1.000000000000&. \end{alignedat} \]
6a (231)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6a}}&{}\approx{}&1.000754750339&{}:{}&-0.000570297495&{}:{}&-0.000184452844&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.064714214022&{}:{}&-0.048898757974&{}:{}&-0.015815456048&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.012753114521&{}:{}&-0.012566450214&{}:{}&-0.000186664307&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.024062650191&{}:{}&-0.000583579907&{}:{}&-0.023479070284&. \end{alignedat} \]
6a (231)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6a}}&{}\approx{}&0.983265606349&{}:{}&0.005574535548&{}:{}&0.011159858103&,\\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.268370108047&{}:{}&0.243719427540&{}:{}&0.487910464413&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.994918993176&{}:{}&-0.006211114771&{}:{}&0.011292121596&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.005900455804&{}:{}&0.005702861783&{}:{}&-0.011603317587&. \end{alignedat} \]
6a (231)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6a}}&{}\approx{}&1.018649834244&{}:{}&-0.006857758045&{}:{}&-0.011792076199&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.018099870087&{}:{}&-0.006355829153&{}:{}&-0.011744040934&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&16.559681942832&{}:{}&15.038620480927&{}:{}&-30.598302423759&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.550039664287&{}:{}&-0.550039664287&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.064714214022&{}:{}&-0.048898757974&{}:{}&-0.015815456048&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.012753114521&{}:{}&-0.012566450214&{}:{}&-0.000186664307&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.024062650191&{}:{}&-0.000583579907&{}:{}&-0.023479070284&, \end{alignedat} \]
6a (231)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6a}}&{}\approx{}&1.111631062982&{}:{}&-0.133409203006&{}:{}&0.021778140024&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.195090322016&{}:{}&-0.195090322016&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.980785280403&{}:{}&0.000000000000&{}:{}&0.019214719597&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.136379290286&{}:{}&-0.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6a}}&{}\approx{}&1.044197063812&{}:{}&-0.024926963029&{}:{}&-0.019270100783&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.064714214022&{}:{}&-0.048898757974&{}:{}&-0.015815456048&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.012753114521&{}:{}&-0.012566450214&{}:{}&-0.000186664307&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.024062650191&{}:{}&-0.000583579907&{}:{}&-0.023479070284&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.195090322016&{}:{}&-0.195090322016&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.980785280403&{}:{}&0.000000000000&{}:{}&0.019214719597&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.136379290286&{}:{}&-0.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)