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 > 4a (211)

4a(211)

Malfatti circles

4a (211)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.541196100146&{}:{}&0.541196100146&{}:{}&1.000000000000&, \\ P_{\mathbf{4a}}&{}\approx{}&1.134180612673&{}:{}&-0.000646332543&{}:{}&-0.133534280131&, \\ P^-_{\mathbf{4a}}&{}\approx{}&0.719801913471&{}:{}&0.133370071477&{}:{}&0.146828015052&, \\ P^+_{\mathbf{4a}}&{}\approx{}&1.954195477811&{}:{}&-0.265851671716&{}:{}&-0.688343806094&, \\ Q_{\mathbf{4a}}&{}\approx{}&2.598494936932&{}:{}&-0.311850891987&{}:{}&-1.286644044945&, \\ I^\prime_{\mathbf{4a}}&{}\approx{}&1.961910428851&{}:{}&-0.046834520438&{}:{}&-0.915075908413&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{4a}}\)
\(P^-_{\mathbf{4a}}\)
\(P^+_{\mathbf{4a}}\)
\(Q_{\mathbf{4a}}\)
\(I^\prime_{\mathbf{4a}}\)
4a (211)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4a}}&{}\approx{}&2.125750467584&{}:{}&-0.395310994322&{}:{}&-0.730439473262&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&6.729173723164&{}:{}&6.704718023924&{}:{}&-12.433891747089&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.911655337330&{}:{}&-0.911655337330&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.605764791221&{}:{}&-0.002917899166&{}:{}&-0.602846892056&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.758529720179&{}:{}&-0.551486858002&{}:{}&-0.207042862176&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.381177196198&{}:{}&-0.001356955228&{}:{}&-1.379820240970&, \\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.323282325216&{}:{}&0.322107426834&{}:{}&0.354610247950&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.982298113268&{}:{}&-0.182671131818&{}:{}&0.200373018550&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.188110838086&{}:{}&0.220141714592&{}:{}&-0.408252552678&, \\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.195090322016&{}:{}&0.804909677984&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.980785280403&{}:{}&0.000000000000&{}:{}&-0.980785280403&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.136379290286&{}:{}&-0.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4a}}}{B^\prime_{\mathbf{4a}}}{C^\prime_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^*_{\mathbf{4a}}}{B^*_{\mathbf{4a}}}{C^*_{\mathbf{4a}}}\)
4a (211)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4a}}}}&{}\approx{}&-0.730439473262&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4a}}}}&{}\approx{}&-12.433891747089&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4a}}}}&{}\approx{}&-1.684519413727&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.541196100146&{}:{}&0.541196100146&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&2.125750467584&{}:{}&-0.395310994322&{}:{}&-0.730439473262&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&6.729173723164&{}:{}&6.704718023924&{}:{}&-12.433891747089&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.911655337330&{}:{}&-0.911655337330&{}:{}&1.000000000000&. \end{alignedat} \]
4a (211)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4a}}&{}\approx{}&1.134180612673&{}:{}&-0.000646332543&{}:{}&-0.133534280131&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.605764791221&{}:{}&-0.002917899166&{}:{}&-0.602846892056&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.758529720179&{}:{}&-0.551486858002&{}:{}&-0.207042862176&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.381177196198&{}:{}&-0.001356955228&{}:{}&-1.379820240970&. \end{alignedat} \]
4a (211)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4a}}&{}\approx{}&0.719801913471&{}:{}&0.133370071477&{}:{}&0.146828015052&,\\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.323282325216&{}:{}&0.322107426834&{}:{}&0.354610247950&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.982298113268&{}:{}&-0.182671131818&{}:{}&0.200373018550&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.188110838086&{}:{}&0.220141714592&{}:{}&-0.408252552678&. \end{alignedat} \]
4a (211)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4a}}&{}\approx{}&1.954195477811&{}:{}&-0.265851671716&{}:{}&-0.688343806094&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&2.125750467584&{}:{}&-0.395310994322&{}:{}&-0.730439473262&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&6.729173723164&{}:{}&6.704718023924&{}:{}&-12.433891747089&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.911655337330&{}:{}&-0.911655337330&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.605764791221&{}:{}&-0.002917899166&{}:{}&-0.602846892056&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.758529720179&{}:{}&-0.551486858002&{}:{}&-0.207042862176&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.381177196198&{}:{}&-0.001356955228&{}:{}&-1.379820240970&, \end{alignedat} \]
4a (211)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4a}}&{}\approx{}&2.598494936932&{}:{}&-0.311850891987&{}:{}&-1.286644044945&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.195090322016&{}:{}&0.804909677984&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.980785280403&{}:{}&0.000000000000&{}:{}&-0.980785280403&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.136379290286&{}:{}&-0.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4a}}&{}\approx{}&1.961910428851&{}:{}&-0.046834520438&{}:{}&-0.915075908413&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.605764791221&{}:{}&-0.002917899166&{}:{}&-0.602846892056&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.758529720179&{}:{}&-0.551486858002&{}:{}&-0.207042862176&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.381177196198&{}:{}&-0.001356955228&{}:{}&-1.379820240970&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.195090322016&{}:{}&0.804909677984&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.980785280403&{}:{}&0.000000000000&{}:{}&-0.980785280403&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.136379290286&{}:{}&-0.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)