Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 108° Top Angle

\(A=36\degree\), \(B=36\degree\), \(C=108\degree\).

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[Top] > Isosceles Triangle with 108° Top Angle > 6c (330)

6c(330)

Malfatti circles

6c (330)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.618033988750&{}:{}&2.618033988750&{}:{}&-4.236067977500&, \\ P_{\mathbf{6c}}&{}\approx{}&0.590470515754&{}:{}&0.590470515754&{}:{}&-0.180941031508&, \\ P^-_{\mathbf{6c}}&{}\approx{}&0.435456545682&{}:{}&0.435456545682&{}:{}&0.129086908636&, \\ P^+_{\mathbf{6c}}&{}\approx{}&0.724925434503&{}:{}&0.724925434503&{}:{}&-0.449850869005&, \\ Q_{\mathbf{6c}}&{}\approx{}&0.421906764831&{}:{}&0.421906764831&{}:{}&0.156186470338&, \\ I^\prime_{\mathbf{6c}}&{}\approx{}&0.771692380705&{}:{}&0.771692380705&{}:{}&-0.543384761410&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{6c}}\)
\(P^-_{\mathbf{6c}}\)
\(P^+_{\mathbf{6c}}\)
\(Q_{\mathbf{6c}}\)
\(I^\prime_{\mathbf{6c}}\)
6c (330)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6c}}&{}\approx{}&1.396802246667&{}:{}&0.642039521920&{}:{}&-1.038841768588&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.642039521920&{}:{}&1.396802246667&{}:{}&-1.038841768588&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.378082272596&{}:{}&0.378082272596&{}:{}&0.243835454809&, \\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.475911333036&{}:{}&0.755644995994&{}:{}&-0.231556329031&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.755644995994&{}:{}&0.475911333036&{}:{}&-0.231556329031&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&1.019420884294&{}:{}&1.019420884294&{}:{}&-1.038841768588&, \\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.261746017743&{}:{}&0.569446915894&{}:{}&0.168807066363&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.569446915894&{}:{}&0.261746017743&{}:{}&0.168807066363&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.796006335767&{}:{}&0.796006335767&{}:{}&-0.592012671535&, \\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.729824774213&{}:{}&0.270175225787&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.729824774213&{}:{}&0.000000000000&{}:{}&0.270175225787&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6c}}}{B^\prime_{\mathbf{6c}}}{C^\prime_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^*_{\mathbf{6c}}}{B^*_{\mathbf{6c}}}{C^*_{\mathbf{6c}}}\)
6c (330)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6c}}}}&{}\approx{}&0.245237275253&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6c}}}}&{}\approx{}&0.245237275253&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6c}}}}&{}\approx{}&0.144414577588&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.618033988750&{}:{}&2.618033988750&{}:{}&-4.236067977500&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.396802246667&{}:{}&0.642039521920&{}:{}&-1.038841768588&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.642039521920&{}:{}&1.396802246667&{}:{}&-1.038841768588&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.378082272596&{}:{}&0.378082272596&{}:{}&0.243835454809&. \end{alignedat} \]
6c (330)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6c}}&{}\approx{}&0.590470515754&{}:{}&0.590470515754&{}:{}&-0.180941031508&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.475911333036&{}:{}&0.755644995994&{}:{}&-0.231556329031&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.755644995994&{}:{}&0.475911333036&{}:{}&-0.231556329031&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&1.019420884294&{}:{}&1.019420884294&{}:{}&-1.038841768588&. \end{alignedat} \]
6c (330)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6c}}&{}\approx{}&0.435456545682&{}:{}&0.435456545682&{}:{}&0.129086908636&,\\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.261746017743&{}:{}&0.569446915894&{}:{}&0.168807066363&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.569446915894&{}:{}&0.261746017743&{}:{}&0.168807066363&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.796006335767&{}:{}&0.796006335767&{}:{}&-0.592012671535&. \end{alignedat} \]
6c (330)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6c}}&{}\approx{}&0.724925434503&{}:{}&0.724925434503&{}:{}&-0.449850869005&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.396802246667&{}:{}&0.642039521920&{}:{}&-1.038841768588&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.642039521920&{}:{}&1.396802246667&{}:{}&-1.038841768588&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.378082272596&{}:{}&0.378082272596&{}:{}&0.243835454809&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.475911333036&{}:{}&0.755644995994&{}:{}&-0.231556329031&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.755644995994&{}:{}&0.475911333036&{}:{}&-0.231556329031&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&1.019420884294&{}:{}&1.019420884294&{}:{}&-1.038841768588&, \end{alignedat} \]
6c (330)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6c}}&{}\approx{}&0.421906764831&{}:{}&0.421906764831&{}:{}&0.156186470338&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.729824774213&{}:{}&0.270175225787&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.729824774213&{}:{}&0.000000000000&{}:{}&0.270175225787&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6c}}&{}\approx{}&0.771692380705&{}:{}&0.771692380705&{}:{}&-0.543384761410&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.475911333036&{}:{}&0.755644995994&{}:{}&-0.231556329031&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.755644995994&{}:{}&0.475911333036&{}:{}&-0.231556329031&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&1.019420884294&{}:{}&1.019420884294&{}:{}&-1.038841768588&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.729824774213&{}:{}&0.270175225787&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.729824774213&{}:{}&0.000000000000&{}:{}&0.270175225787&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)