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 > 1b (103)

1b(103)

Malfatti circles

1b (103)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.541196100146&{}:{}&-0.541196100146&{}:{}&1.000000000000&, \\ P_{\mathbf{1b}}&{}\approx{}&0.863330366494&{}:{}&-0.142559615498&{}:{}&0.279229249005&, \\ P^-_{\mathbf{1b}}&{}\approx{}&0.928595852513&{}:{}&-0.061794522698&{}:{}&0.133198670184&, \\ P^+_{\mathbf{1b}}&{}\approx{}&0.816884897140&{}:{}&-0.200035210506&{}:{}&0.383150313366&, \\ Q_{\mathbf{1b}}&{}\approx{}&1.089815338754&{}:{}&0.088089562205&{}:{}&-0.177904900959&, \\ I^\prime_{\mathbf{1b}}&{}\approx{}&0.731687521071&{}:{}&-0.297327717849&{}:{}&0.565640196777&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{1b}}\)
\(P^-_{\mathbf{1b}}\)
\(P^+_{\mathbf{1b}}\)
\(Q_{\mathbf{1b}}\)
\(I^\prime_{\mathbf{1b}}\)
1b (103)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1b}}&{}\approx{}&0.909005808310&{}:{}&-0.107334967498&{}:{}&0.198329159188&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.437792331916&{}:{}&-0.246727081811&{}:{}&0.808934749895&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&0.823192531266&{}:{}&-0.823192531266&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.571596675585&{}:{}&-0.446866005564&{}:{}&0.875269329979&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.899107374019&{}:{}&-0.189908076237&{}:{}&0.290800702217&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.816224391515&{}:{}&-0.134781121956&{}:{}&0.318556730441&, \\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.605615900531&{}:{}&-0.341307585682&{}:{}&0.735691685151&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.975265862056&{}:{}&-0.115158922691&{}:{}&0.139893060636&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.868329695847&{}:{}&-0.057784039153&{}:{}&0.189454343306&, \\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-0.980785280403&{}:{}&1.980785280403&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.195090322016&{}:{}&0.000000000000&{}:{}&-0.195090322016&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.925215047384&{}:{}&0.074784952616&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1b}}}{B^\prime_{\mathbf{1b}}}{C^\prime_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^*_{\mathbf{1b}}}{B^*_{\mathbf{1b}}}{C^*_{\mathbf{1b}}}\)
1b (103)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1b}}}}&{}\approx{}&0.198329159188&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1b}}}}&{}\approx{}&0.808934749895&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1b}}}}&{}\approx{}&1.521061461906&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.541196100146&{}:{}&-0.541196100146&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.909005808310&{}:{}&-0.107334967498&{}:{}&0.198329159188&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.437792331916&{}:{}&-0.246727081811&{}:{}&0.808934749895&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&0.823192531266&{}:{}&-0.823192531266&{}:{}&1.000000000000&. \end{alignedat} \]
1b (103)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1b}}&{}\approx{}&0.863330366494&{}:{}&-0.142559615498&{}:{}&0.279229249005&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.571596675585&{}:{}&-0.446866005564&{}:{}&0.875269329979&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.899107374019&{}:{}&-0.189908076237&{}:{}&0.290800702217&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.816224391515&{}:{}&-0.134781121956&{}:{}&0.318556730441&. \end{alignedat} \]
1b (103)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1b}}&{}\approx{}&0.928595852513&{}:{}&-0.061794522698&{}:{}&0.133198670184&,\\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.605615900531&{}:{}&-0.341307585682&{}:{}&0.735691685151&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.975265862056&{}:{}&-0.115158922691&{}:{}&0.139893060636&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.868329695847&{}:{}&-0.057784039153&{}:{}&0.189454343306&. \end{alignedat} \]
1b (103)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1b}}&{}\approx{}&0.816884897140&{}:{}&-0.200035210506&{}:{}&0.383150313366&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.909005808310&{}:{}&-0.107334967498&{}:{}&0.198329159188&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.437792331916&{}:{}&-0.246727081811&{}:{}&0.808934749895&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&0.823192531266&{}:{}&-0.823192531266&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.571596675585&{}:{}&-0.446866005564&{}:{}&0.875269329979&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.899107374019&{}:{}&-0.189908076237&{}:{}&0.290800702217&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.816224391515&{}:{}&-0.134781121956&{}:{}&0.318556730441&, \end{alignedat} \]
1b (103)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1b}}&{}\approx{}&1.089815338754&{}:{}&0.088089562205&{}:{}&-0.177904900959&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-0.980785280403&{}:{}&1.980785280403&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.195090322016&{}:{}&0.000000000000&{}:{}&-0.195090322016&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.925215047384&{}:{}&0.074784952616&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1b}}&{}\approx{}&0.731687521071&{}:{}&-0.297327717849&{}:{}&0.565640196777&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.571596675585&{}:{}&-0.446866005564&{}:{}&0.875269329979&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.899107374019&{}:{}&-0.189908076237&{}:{}&0.290800702217&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.816224391515&{}:{}&-0.134781121956&{}:{}&0.318556730441&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-0.980785280403&{}:{}&1.980785280403&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.195090322016&{}:{}&0.000000000000&{}:{}&-0.195090322016&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.925215047384&{}:{}&0.074784952616&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)