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 > 2b (121)

2b(121)

Malfatti circles

2b (121)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.541196100146&{}:{}&-0.541196100146&{}:{}&1.000000000000&, \\ P_{\mathbf{2b}}&{}\approx{}&-0.000646332543&{}:{}&1.134180612673&{}:{}&-0.133534280131&, \\ P^-_{\mathbf{2b}}&{}\approx{}&0.133370071477&{}:{}&0.719801913471&{}:{}&0.146828015052&, \\ P^+_{\mathbf{2b}}&{}\approx{}&-0.265851671716&{}:{}&1.954195477811&{}:{}&-0.688343806094&, \\ Q_{\mathbf{2b}}&{}\approx{}&-0.311850891987&{}:{}&2.598494936932&{}:{}&-1.286644044945&, \\ I^\prime_{\mathbf{2b}}&{}\approx{}&-0.046834520438&{}:{}&1.961910428851&{}:{}&-0.915075908413&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{2b}}\)
\(P^-_{\mathbf{2b}}\)
\(P^+_{\mathbf{2b}}\)
\(Q_{\mathbf{2b}}\)
\(I^\prime_{\mathbf{2b}}\)
2b (121)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2b}}&{}\approx{}&6.704718023924&{}:{}&6.729173723164&{}:{}&-12.433891747089&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.395310994322&{}:{}&2.125750467584&{}:{}&-0.730439473262&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.911655337330&{}:{}&0.911655337330&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.551486858002&{}:{}&1.758529720179&{}:{}&-0.207042862176&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.002917899166&{}:{}&1.605764791221&{}:{}&-0.602846892056&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.001356955228&{}:{}&2.381177196198&{}:{}&-1.379820240970&, \\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.182671131818&{}:{}&0.982298113268&{}:{}&0.200373018550&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.322107426834&{}:{}&0.323282325216&{}:{}&0.354610247950&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.220141714592&{}:{}&1.188110838086&{}:{}&-0.408252552678&, \\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&1.980785280403&{}:{}&-0.980785280403&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.195090322016&{}:{}&0.000000000000&{}:{}&0.804909677984&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.136379290286&{}:{}&1.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2b}}}{B^\prime_{\mathbf{2b}}}{C^\prime_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^*_{\mathbf{2b}}}{B^*_{\mathbf{2b}}}{C^*_{\mathbf{2b}}}\)
2b (121)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2b}}}}&{}\approx{}&-12.433891747089&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2b}}}}&{}\approx{}&-0.730439473262&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2b}}}}&{}\approx{}&-1.684519413727&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.541196100146&{}:{}&-0.541196100146&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&6.704718023924&{}:{}&6.729173723164&{}:{}&-12.433891747089&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.395310994322&{}:{}&2.125750467584&{}:{}&-0.730439473262&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.911655337330&{}:{}&0.911655337330&{}:{}&1.000000000000&. \end{alignedat} \]
2b (121)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2b}}&{}\approx{}&-0.000646332543&{}:{}&1.134180612673&{}:{}&-0.133534280131&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.551486858002&{}:{}&1.758529720179&{}:{}&-0.207042862176&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.002917899166&{}:{}&1.605764791221&{}:{}&-0.602846892056&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.001356955228&{}:{}&2.381177196198&{}:{}&-1.379820240970&. \end{alignedat} \]
2b (121)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2b}}&{}\approx{}&0.133370071477&{}:{}&0.719801913471&{}:{}&0.146828015052&,\\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.182671131818&{}:{}&0.982298113268&{}:{}&0.200373018550&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.322107426834&{}:{}&0.323282325216&{}:{}&0.354610247950&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.220141714592&{}:{}&1.188110838086&{}:{}&-0.408252552678&. \end{alignedat} \]
2b (121)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2b}}&{}\approx{}&-0.265851671716&{}:{}&1.954195477811&{}:{}&-0.688343806094&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&6.704718023924&{}:{}&6.729173723164&{}:{}&-12.433891747089&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.395310994322&{}:{}&2.125750467584&{}:{}&-0.730439473262&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.911655337330&{}:{}&0.911655337330&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.551486858002&{}:{}&1.758529720179&{}:{}&-0.207042862176&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.002917899166&{}:{}&1.605764791221&{}:{}&-0.602846892056&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.001356955228&{}:{}&2.381177196198&{}:{}&-1.379820240970&, \end{alignedat} \]
2b (121)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2b}}&{}\approx{}&-0.311850891987&{}:{}&2.598494936932&{}:{}&-1.286644044945&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&1.980785280403&{}:{}&-0.980785280403&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.195090322016&{}:{}&0.000000000000&{}:{}&0.804909677984&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.136379290286&{}:{}&1.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2b}}&{}\approx{}&-0.046834520438&{}:{}&1.961910428851&{}:{}&-0.915075908413&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.551486858002&{}:{}&1.758529720179&{}:{}&-0.207042862176&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.002917899166&{}:{}&1.605764791221&{}:{}&-0.602846892056&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.001356955228&{}:{}&2.381177196198&{}:{}&-1.379820240970&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&1.980785280403&{}:{}&-0.980785280403&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.195090322016&{}:{}&0.000000000000&{}:{}&0.804909677984&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.136379290286&{}:{}&1.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)