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 > 7b (323)

7b(323)

Malfatti circles

7b (323)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.541196100146&{}:{}&-0.541196100146&{}:{}&1.000000000000&, \\ P_{\mathbf{7b}}&{}\approx{}&-0.000258618149&{}:{}&1.000443013535&{}:{}&-0.000184395386&, \\ P^-_{\mathbf{7b}}&{}\approx{}&0.003891720452&{}:{}&0.988626098647&{}:{}&0.007482180901&, \\ P^+_{\mathbf{7b}}&{}\approx{}&-0.004473573387&{}:{}&1.012443906015&{}:{}&-0.007970332628&, \\ Q_{\mathbf{7b}}&{}\approx{}&0.073453531382&{}:{}&0.908743138030&{}:{}&0.017803330588&, \\ I^\prime_{\mathbf{7b}}&{}\approx{}&-0.016653122231&{}:{}&1.035767662910&{}:{}&-0.019114540678&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{7b}}\)
\(P^-_{\mathbf{7b}}\)
\(P^+_{\mathbf{7b}}\)
\(Q_{\mathbf{7b}}\)
\(I^\prime_{\mathbf{7b}}\)
7b (323)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7b}}&{}\approx{}&10.380306311212&{}:{}&11.064825724938&{}:{}&-20.445132036150&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.004246829266&{}:{}&1.012093946541&{}:{}&-0.007847117275&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.554431429201&{}:{}&0.554431429201&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.008429093433&{}:{}&1.008614995028&{}:{}&-0.000185901595&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.032662938072&{}:{}&1.055951694529&{}:{}&-0.023288756457&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.000262626978&{}:{}&1.015950840180&{}:{}&-0.015688213202&, \\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.004181979688&{}:{}&0.996639153889&{}:{}&0.007542825799&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.250718713165&{}:{}&0.267252119926&{}:{}&0.482029166909&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.003951576337&{}:{}&1.003831479147&{}:{}&-0.007783055484&, \\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.980785280403&{}:{}&0.019214719597&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.804909677984&{}:{}&0.000000000000&{}:{}&0.195090322016&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.074784952616&{}:{}&0.925215047384&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7b}}}{B^\prime_{\mathbf{7b}}}{C^\prime_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^*_{\mathbf{7b}}}{B^*_{\mathbf{7b}}}{C^*_{\mathbf{7b}}}\)
7b (323)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7b}}}}&{}\approx{}&-20.445132036150&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7b}}}}&{}\approx{}&-0.007847117275&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7b}}}}&{}\approx{}&-1.024455699240&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.541196100146&{}:{}&-0.541196100146&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&10.380306311212&{}:{}&11.064825724938&{}:{}&-20.445132036150&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.004246829266&{}:{}&1.012093946541&{}:{}&-0.007847117275&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.554431429201&{}:{}&0.554431429201&{}:{}&1.000000000000&. \end{alignedat} \]
7b (323)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7b}}&{}\approx{}&-0.000258618149&{}:{}&1.000443013535&{}:{}&-0.000184395386&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.008429093433&{}:{}&1.008614995028&{}:{}&-0.000185901595&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.032662938072&{}:{}&1.055951694529&{}:{}&-0.023288756457&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.000262626978&{}:{}&1.015950840180&{}:{}&-0.015688213202&. \end{alignedat} \]
7b (323)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7b}}&{}\approx{}&0.003891720452&{}:{}&0.988626098647&{}:{}&0.007482180901&,\\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.004181979688&{}:{}&0.996639153889&{}:{}&0.007542825799&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.250718713165&{}:{}&0.267252119926&{}:{}&0.482029166909&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.003951576337&{}:{}&1.003831479147&{}:{}&-0.007783055484&. \end{alignedat} \]
7b (323)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7b}}&{}\approx{}&-0.004473573387&{}:{}&1.012443906015&{}:{}&-0.007970332628&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&10.380306311212&{}:{}&11.064825724938&{}:{}&-20.445132036150&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.004246829266&{}:{}&1.012093946541&{}:{}&-0.007847117275&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.554431429201&{}:{}&0.554431429201&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.008429093433&{}:{}&1.008614995028&{}:{}&-0.000185901595&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.032662938072&{}:{}&1.055951694529&{}:{}&-0.023288756457&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.000262626978&{}:{}&1.015950840180&{}:{}&-0.015688213202&, \end{alignedat} \]
7b (323)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7b}}&{}\approx{}&0.073453531382&{}:{}&0.908743138030&{}:{}&0.017803330588&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.980785280403&{}:{}&0.019214719597&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.804909677984&{}:{}&0.000000000000&{}:{}&0.195090322016&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.074784952616&{}:{}&0.925215047384&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7b}}&{}\approx{}&-0.016653122231&{}:{}&1.035767662910&{}:{}&-0.019114540678&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.008429093433&{}:{}&1.008614995028&{}:{}&-0.000185901595&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.032662938072&{}:{}&1.055951694529&{}:{}&-0.023288756457&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.000262626978&{}:{}&1.015950840180&{}:{}&-0.015688213202&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.980785280403&{}:{}&0.019214719597&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.804909677984&{}:{}&0.000000000000&{}:{}&0.195090322016&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.074784952616&{}:{}&0.925215047384&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)