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 > 4b (301)

4b(301)

Malfatti circles

4b (301)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.618033988750&{}:{}&-0.618033988750&{}:{}&1.000000000000&, \\ P_{\mathbf{4b}}&{}\approx{}&0.013061137105&{}:{}&-0.005879384437&{}:{}&0.992818247331&, \\ P^-_{\mathbf{4b}}&{}\approx{}&-0.010261814045&{}:{}&0.017720438091&{}:{}&0.992541375954&, \\ P^+_{\mathbf{4b}}&{}\approx{}&0.034714522894&{}:{}&-0.027789821861&{}:{}&0.993075298967&, \\ Q_{\mathbf{4b}}&{}\approx{}&-0.289513911147&{}:{}&0.063113303097&{}:{}&1.226400608049&, \\ I^\prime_{\mathbf{4b}}&{}\approx{}&0.087571481518&{}:{}&-0.058754093781&{}:{}&0.971182612263&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{4b}}\)
\(P^-_{\mathbf{4b}}\)
\(P^+_{\mathbf{4b}}\)
\(Q_{\mathbf{4b}}\)
\(I^\prime_{\mathbf{4b}}\)
4b (301)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4b}}&{}\approx{}&-2.298915279258&{}:{}&-5.337757047845&{}:{}&8.636672327103&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.297463005165&{}:{}&0.221231742082&{}:{}&0.481305252753&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.024005533170&{}:{}&-0.024005533170&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.044425859367&{}:{}&-0.005692538760&{}:{}&0.961266679393&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.013605417206&{}:{}&-0.047796112540&{}:{}&1.034190695334&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.160409530808&{}:{}&-0.072207288794&{}:{}&0.911797757986&, \\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.023041054567&{}:{}&0.017136291077&{}:{}&0.959822654356&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.010706642079&{}:{}&-0.024859312882&{}:{}&1.035565954961&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.223031100575&{}:{}&0.385137441858&{}:{}&0.837893658717&, \\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.048943483705&{}:{}&0.951056516295&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.309016994375&{}:{}&0.000000000000&{}:{}&1.309016994375&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.278768257918&{}:{}&-0.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4b}}}{B^\prime_{\mathbf{4b}}}{C^\prime_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^*_{\mathbf{4b}}}{B^*_{\mathbf{4b}}}{C^*_{\mathbf{4b}}}\)
4b (301)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4b}}}}&{}\approx{}&8.636672327103&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4b}}}}&{}\approx{}&0.481305252753&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4b}}}}&{}\approx{}&0.038841768588&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.618033988750&{}:{}&-0.618033988750&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&-2.298915279258&{}:{}&-5.337757047845&{}:{}&8.636672327103&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.297463005165&{}:{}&0.221231742082&{}:{}&0.481305252753&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.024005533170&{}:{}&-0.024005533170&{}:{}&1.000000000000&. \end{alignedat} \]
4b (301)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4b}}&{}\approx{}&0.013061137105&{}:{}&-0.005879384437&{}:{}&0.992818247331&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.044425859367&{}:{}&-0.005692538760&{}:{}&0.961266679393&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.013605417206&{}:{}&-0.047796112540&{}:{}&1.034190695334&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.160409530808&{}:{}&-0.072207288794&{}:{}&0.911797757986&. \end{alignedat} \]
4b (301)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4b}}&{}\approx{}&-0.010261814045&{}:{}&0.017720438091&{}:{}&0.992541375954&,\\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.023041054567&{}:{}&0.017136291077&{}:{}&0.959822654356&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.010706642079&{}:{}&-0.024859312882&{}:{}&1.035565954961&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.223031100575&{}:{}&0.385137441858&{}:{}&0.837893658717&. \end{alignedat} \]
4b (301)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4b}}&{}\approx{}&0.034714522894&{}:{}&-0.027789821861&{}:{}&0.993075298967&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&-2.298915279258&{}:{}&-5.337757047845&{}:{}&8.636672327103&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.297463005165&{}:{}&0.221231742082&{}:{}&0.481305252753&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.024005533170&{}:{}&-0.024005533170&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.044425859367&{}:{}&-0.005692538760&{}:{}&0.961266679393&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.013605417206&{}:{}&-0.047796112540&{}:{}&1.034190695334&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.160409530808&{}:{}&-0.072207288794&{}:{}&0.911797757986&, \end{alignedat} \]
4b (301)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4b}}&{}\approx{}&-0.289513911147&{}:{}&0.063113303097&{}:{}&1.226400608049&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.048943483705&{}:{}&0.951056516295&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.309016994375&{}:{}&0.000000000000&{}:{}&1.309016994375&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.278768257918&{}:{}&-0.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4b}}&{}\approx{}&0.087571481518&{}:{}&-0.058754093781&{}:{}&0.971182612263&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.044425859367&{}:{}&-0.005692538760&{}:{}&0.961266679393&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.013605417206&{}:{}&-0.047796112540&{}:{}&1.034190695334&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.160409530808&{}:{}&-0.072207288794&{}:{}&0.911797757986&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.048943483705&{}:{}&0.951056516295&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.309016994375&{}:{}&0.000000000000&{}:{}&1.309016994375&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.278768257918&{}:{}&-0.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)