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 > 0c (110)

0c(110)

Malfatti circles

0c (110)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.618033988750&{}:{}&2.618033988750&{}:{}&-4.236067977500&, \\ P_{\mathbf{c}}&{}\approx{}&1.110804778477&{}:{}&1.110804778477&{}:{}&-1.221609556953&, \\ P^-_{\mathbf{c}}&{}\approx{}&0.878366934585&{}:{}&0.878366934585&{}:{}&-0.756733869169&, \\ P^+_{\mathbf{c}}&{}\approx{}&1.288451063029&{}:{}&1.288451063029&{}:{}&-1.576902126058&, \\ Q_{\mathbf{c}}&{}\approx{}&0.770001728548&{}:{}&0.770001728548&{}:{}&-0.540003457095&, \\ I^\prime_{\mathbf{c}}&{}\approx{}&1.501399931367&{}:{}&1.501399931367&{}:{}&-2.002799862733&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{c}}\)
\(P^-_{\mathbf{c}}\)
\(P^+_{\mathbf{c}}\)
\(Q_{\mathbf{c}}\)
\(I^\prime_{\mathbf{c}}\)
0c (110)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{c}}&{}\approx{}&1.396802246667&{}:{}&0.642039521920&{}:{}&-1.038841768588&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.642039521920&{}:{}&1.396802246667&{}:{}&-1.038841768588&,\\C^\prime_{\mathbf{c}}&{}\approx{}&7.955791036595&{}:{}&7.955791036595&{}:{}&-14.911582073190&, \\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.188190960236&{}:{}&1.886592083569&{}:{}&-2.074783043805&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.886592083569&{}:{}&1.188190960236&{}:{}&-2.074783043805&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.019420884294&{}:{}&1.019420884294&{}:{}&-1.038841768588&, \\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.768482912340&{}:{}&1.671888757365&{}:{}&-1.440371669705&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.671888757365&{}:{}&0.768482912340&{}:{}&-1.440371669705&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.796006335767&{}:{}&0.796006335767&{}:{}&-0.592012671535&, \\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&3.347858762963&{}:{}&-2.347858762963&,\\B^*_{\mathbf{c}}&{}\approx{}&3.347858762963&{}:{}&0.000000000000&{}:{}&-2.347858762963&,\\C^*_{\mathbf{c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{c}}}{B^\prime_{\mathbf{c}}}{C^\prime_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{c}}}{B^{\prime\prime}_{\mathbf{c}}}{C^{\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{c}}}{B^{\prime\prime\prime}_{\mathbf{c}}}{C^{\prime\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^*_{\mathbf{c}}}{B^*_{\mathbf{c}}}{C^*_{\mathbf{c}}}\)
0c (110)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{c}}}}&{}\approx{}&0.245237275253&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{c}}}}&{}\approx{}&0.245237275253&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{c}}}}&{}\approx{}&3.038841768588&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.618033988750&{}:{}&2.618033988750&{}:{}&-4.236067977500&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.396802246667&{}:{}&0.642039521920&{}:{}&-1.038841768588&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.642039521920&{}:{}&1.396802246667&{}:{}&-1.038841768588&,\\C^\prime_{\mathbf{c}}&{}\approx{}&7.955791036595&{}:{}&7.955791036595&{}:{}&-14.911582073190&. \end{alignedat} \]
0c (110)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{c}}&{}\approx{}&1.110804778477&{}:{}&1.110804778477&{}:{}&-1.221609556953&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.188190960236&{}:{}&1.886592083569&{}:{}&-2.074783043805&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.886592083569&{}:{}&1.188190960236&{}:{}&-2.074783043805&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.019420884294&{}:{}&1.019420884294&{}:{}&-1.038841768588&. \end{alignedat} \]
0c (110)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{c}}&{}\approx{}&0.878366934585&{}:{}&0.878366934585&{}:{}&-0.756733869169&,\\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.768482912340&{}:{}&1.671888757365&{}:{}&-1.440371669705&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.671888757365&{}:{}&0.768482912340&{}:{}&-1.440371669705&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.796006335767&{}:{}&0.796006335767&{}:{}&-0.592012671535&. \end{alignedat} \]
0c (110)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{c}}&{}\approx{}&1.288451063029&{}:{}&1.288451063029&{}:{}&-1.576902126058&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.396802246667&{}:{}&0.642039521920&{}:{}&-1.038841768588&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.642039521920&{}:{}&1.396802246667&{}:{}&-1.038841768588&,\\C^\prime_{\mathbf{c}}&{}\approx{}&7.955791036595&{}:{}&7.955791036595&{}:{}&-14.911582073190&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.188190960236&{}:{}&1.886592083569&{}:{}&-2.074783043805&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.886592083569&{}:{}&1.188190960236&{}:{}&-2.074783043805&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.019420884294&{}:{}&1.019420884294&{}:{}&-1.038841768588&, \end{alignedat} \]
0c (110)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{c}}&{}\approx{}&0.770001728548&{}:{}&0.770001728548&{}:{}&-0.540003457095&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&3.347858762963&{}:{}&-2.347858762963&,\\B^*_{\mathbf{c}}&{}\approx{}&3.347858762963&{}:{}&0.000000000000&{}:{}&-2.347858762963&,\\C^*_{\mathbf{c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{c}}&{}\approx{}&1.501399931367&{}:{}&1.501399931367&{}:{}&-2.002799862733&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.188190960236&{}:{}&1.886592083569&{}:{}&-2.074783043805&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.886592083569&{}:{}&1.188190960236&{}:{}&-2.074783043805&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.019420884294&{}:{}&1.019420884294&{}:{}&-1.038841768588&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&3.347858762963&{}:{}&-2.347858762963&,\\B^*_{\mathbf{c}}&{}\approx{}&3.347858762963&{}:{}&0.000000000000&{}:{}&-2.347858762963&,\\C^*_{\mathbf{c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)