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 > 7a (233)

7a(233)

Malfatti circles

7a (233)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.541196100146&{}:{}&0.541196100146&{}:{}&1.000000000000&, \\ P_{\mathbf{7a}}&{}\approx{}&1.000443013535&{}:{}&-0.000258618149&{}:{}&-0.000184395386&, \\ P^-_{\mathbf{7a}}&{}\approx{}&0.988626098647&{}:{}&0.003891720452&{}:{}&0.007482180901&, \\ P^+_{\mathbf{7a}}&{}\approx{}&1.012443906015&{}:{}&-0.004473573387&{}:{}&-0.007970332628&, \\ Q_{\mathbf{7a}}&{}\approx{}&0.908743138030&{}:{}&0.073453531382&{}:{}&0.017803330588&, \\ I^\prime_{\mathbf{7a}}&{}\approx{}&1.035767662910&{}:{}&-0.016653122231&{}:{}&-0.019114540678&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{7a}}\)
\(P^-_{\mathbf{7a}}\)
\(P^+_{\mathbf{7a}}\)
\(Q_{\mathbf{7a}}\)
\(I^\prime_{\mathbf{7a}}\)
7a (233)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7a}}&{}\approx{}&1.012093946541&{}:{}&-0.004246829266&{}:{}&-0.007847117275&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&11.064825724938&{}:{}&10.380306311212&{}:{}&-20.445132036150&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.554431429201&{}:{}&-0.554431429201&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.055951694529&{}:{}&-0.032662938072&{}:{}&-0.023288756457&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.008614995028&{}:{}&-0.008429093433&{}:{}&-0.000185901595&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.015950840180&{}:{}&-0.000262626978&{}:{}&-0.015688213202&, \\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.267252119926&{}:{}&0.250718713165&{}:{}&0.482029166909&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.996639153889&{}:{}&-0.004181979688&{}:{}&0.007542825799&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.003831479147&{}:{}&0.003951576337&{}:{}&-0.007783055484&, \\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.804909677984&{}:{}&0.195090322016&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.980785280403&{}:{}&0.000000000000&{}:{}&0.019214719597&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.925215047384&{}:{}&0.074784952616&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7a}}}{B^\prime_{\mathbf{7a}}}{C^\prime_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^*_{\mathbf{7a}}}{B^*_{\mathbf{7a}}}{C^*_{\mathbf{7a}}}\)
7a (233)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7a}}}}&{}\approx{}&-0.007847117275&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7a}}}}&{}\approx{}&-20.445132036150&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7a}}}}&{}\approx{}&-1.024455699240&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.541196100146&{}:{}&0.541196100146&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.012093946541&{}:{}&-0.004246829266&{}:{}&-0.007847117275&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&11.064825724938&{}:{}&10.380306311212&{}:{}&-20.445132036150&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.554431429201&{}:{}&-0.554431429201&{}:{}&1.000000000000&. \end{alignedat} \]
7a (233)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7a}}&{}\approx{}&1.000443013535&{}:{}&-0.000258618149&{}:{}&-0.000184395386&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.055951694529&{}:{}&-0.032662938072&{}:{}&-0.023288756457&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.008614995028&{}:{}&-0.008429093433&{}:{}&-0.000185901595&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.015950840180&{}:{}&-0.000262626978&{}:{}&-0.015688213202&. \end{alignedat} \]
7a (233)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7a}}&{}\approx{}&0.988626098647&{}:{}&0.003891720452&{}:{}&0.007482180901&,\\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.267252119926&{}:{}&0.250718713165&{}:{}&0.482029166909&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.996639153889&{}:{}&-0.004181979688&{}:{}&0.007542825799&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.003831479147&{}:{}&0.003951576337&{}:{}&-0.007783055484&. \end{alignedat} \]
7a (233)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7a}}&{}\approx{}&1.012443906015&{}:{}&-0.004473573387&{}:{}&-0.007970332628&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.012093946541&{}:{}&-0.004246829266&{}:{}&-0.007847117275&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&11.064825724938&{}:{}&10.380306311212&{}:{}&-20.445132036150&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.554431429201&{}:{}&-0.554431429201&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.055951694529&{}:{}&-0.032662938072&{}:{}&-0.023288756457&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.008614995028&{}:{}&-0.008429093433&{}:{}&-0.000185901595&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.015950840180&{}:{}&-0.000262626978&{}:{}&-0.015688213202&, \end{alignedat} \]
7a (233)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7a}}&{}\approx{}&0.908743138030&{}:{}&0.073453531382&{}:{}&0.017803330588&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.804909677984&{}:{}&0.195090322016&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.980785280403&{}:{}&0.000000000000&{}:{}&0.019214719597&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.925215047384&{}:{}&0.074784952616&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7a}}&{}\approx{}&1.035767662910&{}:{}&-0.016653122231&{}:{}&-0.019114540678&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.055951694529&{}:{}&-0.032662938072&{}:{}&-0.023288756457&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.008614995028&{}:{}&-0.008429093433&{}:{}&-0.000185901595&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.015950840180&{}:{}&-0.000262626978&{}:{}&-0.015688213202&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.804909677984&{}:{}&0.195090322016&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.980785280403&{}:{}&0.000000000000&{}:{}&0.019214719597&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.925215047384&{}:{}&0.074784952616&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)