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 > 2a (031)

2a(031)

Malfatti circles

2a (031)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.618033988750&{}:{}&0.618033988750&{}:{}&1.000000000000&, \\ P_{\mathbf{2a}}&{}\approx{}&-0.005879384437&{}:{}&0.013061137105&{}:{}&0.992818247331&, \\ P^-_{\mathbf{2a}}&{}\approx{}&0.017720438091&{}:{}&-0.010261814045&{}:{}&0.992541375954&, \\ P^+_{\mathbf{2a}}&{}\approx{}&-0.027789821861&{}:{}&0.034714522894&{}:{}&0.993075298967&, \\ Q_{\mathbf{2a}}&{}\approx{}&0.063113303097&{}:{}&-0.289513911147&{}:{}&1.226400608049&, \\ I^\prime_{\mathbf{2a}}&{}\approx{}&-0.058754093781&{}:{}&0.087571481518&{}:{}&0.971182612263&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{2a}}\)
\(P^-_{\mathbf{2a}}\)
\(P^+_{\mathbf{2a}}\)
\(Q_{\mathbf{2a}}\)
\(I^\prime_{\mathbf{2a}}\)
2a (031)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2a}}&{}\approx{}&0.221231742082&{}:{}&0.297463005165&{}:{}&0.481305252753&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-5.337757047845&{}:{}&-2.298915279258&{}:{}&8.636672327103&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.024005533170&{}:{}&0.024005533170&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.047796112540&{}:{}&0.013605417206&{}:{}&1.034190695334&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.005692538760&{}:{}&0.044425859367&{}:{}&0.961266679393&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.072207288794&{}:{}&0.160409530808&{}:{}&0.911797757986&, \\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.024859312882&{}:{}&-0.010706642079&{}:{}&1.035565954961&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.017136291077&{}:{}&0.023041054567&{}:{}&0.959822654356&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.385137441858&{}:{}&-0.223031100575&{}:{}&0.837893658717&, \\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.309016994375&{}:{}&1.309016994375&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.048943483705&{}:{}&0.000000000000&{}:{}&0.951056516295&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.278768257918&{}:{}&1.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2a}}}{B^\prime_{\mathbf{2a}}}{C^\prime_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^*_{\mathbf{2a}}}{B^*_{\mathbf{2a}}}{C^*_{\mathbf{2a}}}\)
2a (031)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2a}}}}&{}\approx{}&0.481305252753&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2a}}}}&{}\approx{}&8.636672327103&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2a}}}}&{}\approx{}&0.038841768588&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.618033988750&{}:{}&0.618033988750&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.221231742082&{}:{}&0.297463005165&{}:{}&0.481305252753&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-5.337757047845&{}:{}&-2.298915279258&{}:{}&8.636672327103&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.024005533170&{}:{}&0.024005533170&{}:{}&1.000000000000&. \end{alignedat} \]
2a (031)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2a}}&{}\approx{}&-0.005879384437&{}:{}&0.013061137105&{}:{}&0.992818247331&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.047796112540&{}:{}&0.013605417206&{}:{}&1.034190695334&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.005692538760&{}:{}&0.044425859367&{}:{}&0.961266679393&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.072207288794&{}:{}&0.160409530808&{}:{}&0.911797757986&. \end{alignedat} \]
2a (031)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2a}}&{}\approx{}&0.017720438091&{}:{}&-0.010261814045&{}:{}&0.992541375954&,\\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.024859312882&{}:{}&-0.010706642079&{}:{}&1.035565954961&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.017136291077&{}:{}&0.023041054567&{}:{}&0.959822654356&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.385137441858&{}:{}&-0.223031100575&{}:{}&0.837893658717&. \end{alignedat} \]
2a (031)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2a}}&{}\approx{}&-0.027789821861&{}:{}&0.034714522894&{}:{}&0.993075298967&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.221231742082&{}:{}&0.297463005165&{}:{}&0.481305252753&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-5.337757047845&{}:{}&-2.298915279258&{}:{}&8.636672327103&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.024005533170&{}:{}&0.024005533170&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.047796112540&{}:{}&0.013605417206&{}:{}&1.034190695334&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.005692538760&{}:{}&0.044425859367&{}:{}&0.961266679393&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.072207288794&{}:{}&0.160409530808&{}:{}&0.911797757986&, \end{alignedat} \]
2a (031)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2a}}&{}\approx{}&0.063113303097&{}:{}&-0.289513911147&{}:{}&1.226400608049&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.309016994375&{}:{}&1.309016994375&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.048943483705&{}:{}&0.000000000000&{}:{}&0.951056516295&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.278768257918&{}:{}&1.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2a}}&{}\approx{}&-0.058754093781&{}:{}&0.087571481518&{}:{}&0.971182612263&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.047796112540&{}:{}&0.013605417206&{}:{}&1.034190695334&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.005692538760&{}:{}&0.044425859367&{}:{}&0.961266679393&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.072207288794&{}:{}&0.160409530808&{}:{}&0.911797757986&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.309016994375&{}:{}&1.309016994375&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.048943483705&{}:{}&0.000000000000&{}:{}&0.951056516295&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.278768257918&{}:{}&1.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)