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 > 1a (013)

1a(013)

Malfatti circles

1a (013)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.541196100146&{}:{}&0.541196100146&{}:{}&1.000000000000&, \\ P_{\mathbf{1a}}&{}\approx{}&-0.142559615498&{}:{}&0.863330366494&{}:{}&0.279229249005&, \\ P^-_{\mathbf{1a}}&{}\approx{}&-0.061794522698&{}:{}&0.928595852513&{}:{}&0.133198670184&, \\ P^+_{\mathbf{1a}}&{}\approx{}&-0.200035210506&{}:{}&0.816884897140&{}:{}&0.383150313366&, \\ Q_{\mathbf{1a}}&{}\approx{}&0.088089562205&{}:{}&1.089815338754&{}:{}&-0.177904900959&, \\ I^\prime_{\mathbf{1a}}&{}\approx{}&-0.297327717849&{}:{}&0.731687521071&{}:{}&0.565640196777&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{1a}}\)
\(P^-_{\mathbf{1a}}\)
\(P^+_{\mathbf{1a}}\)
\(Q_{\mathbf{1a}}\)
\(I^\prime_{\mathbf{1a}}\)
1a (013)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1a}}&{}\approx{}&-0.246727081811&{}:{}&0.437792331916&{}:{}&0.808934749895&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.107334967498&{}:{}&0.909005808310&{}:{}&0.198329159188&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-0.823192531266&{}:{}&0.823192531266&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.189908076237&{}:{}&0.899107374019&{}:{}&0.290800702217&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.446866005564&{}:{}&0.571596675585&{}:{}&0.875269329979&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.134781121956&{}:{}&0.816224391515&{}:{}&0.318556730441&, \\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.115158922691&{}:{}&0.975265862056&{}:{}&0.139893060636&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.341307585682&{}:{}&0.605615900531&{}:{}&0.735691685151&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.057784039153&{}:{}&0.868329695847&{}:{}&0.189454343306&, \\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.195090322016&{}:{}&-0.195090322016&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.980785280403&{}:{}&0.000000000000&{}:{}&1.980785280403&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.074784952616&{}:{}&0.925215047384&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1a}}}{B^\prime_{\mathbf{1a}}}{C^\prime_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^*_{\mathbf{1a}}}{B^*_{\mathbf{1a}}}{C^*_{\mathbf{1a}}}\)
1a (013)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1a}}}}&{}\approx{}&0.808934749895&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1a}}}}&{}\approx{}&0.198329159188&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1a}}}}&{}\approx{}&1.521061461906&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.541196100146&{}:{}&0.541196100146&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&-0.246727081811&{}:{}&0.437792331916&{}:{}&0.808934749895&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.107334967498&{}:{}&0.909005808310&{}:{}&0.198329159188&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-0.823192531266&{}:{}&0.823192531266&{}:{}&1.000000000000&. \end{alignedat} \]
1a (013)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1a}}&{}\approx{}&-0.142559615498&{}:{}&0.863330366494&{}:{}&0.279229249005&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.189908076237&{}:{}&0.899107374019&{}:{}&0.290800702217&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.446866005564&{}:{}&0.571596675585&{}:{}&0.875269329979&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.134781121956&{}:{}&0.816224391515&{}:{}&0.318556730441&. \end{alignedat} \]
1a (013)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1a}}&{}\approx{}&-0.061794522698&{}:{}&0.928595852513&{}:{}&0.133198670184&,\\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.115158922691&{}:{}&0.975265862056&{}:{}&0.139893060636&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.341307585682&{}:{}&0.605615900531&{}:{}&0.735691685151&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.057784039153&{}:{}&0.868329695847&{}:{}&0.189454343306&. \end{alignedat} \]
1a (013)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1a}}&{}\approx{}&-0.200035210506&{}:{}&0.816884897140&{}:{}&0.383150313366&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&-0.246727081811&{}:{}&0.437792331916&{}:{}&0.808934749895&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.107334967498&{}:{}&0.909005808310&{}:{}&0.198329159188&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-0.823192531266&{}:{}&0.823192531266&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.189908076237&{}:{}&0.899107374019&{}:{}&0.290800702217&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.446866005564&{}:{}&0.571596675585&{}:{}&0.875269329979&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.134781121956&{}:{}&0.816224391515&{}:{}&0.318556730441&, \end{alignedat} \]
1a (013)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1a}}&{}\approx{}&0.088089562205&{}:{}&1.089815338754&{}:{}&-0.177904900959&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.195090322016&{}:{}&-0.195090322016&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.980785280403&{}:{}&0.000000000000&{}:{}&1.980785280403&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.074784952616&{}:{}&0.925215047384&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1a}}&{}\approx{}&-0.297327717849&{}:{}&0.731687521071&{}:{}&0.565640196777&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.189908076237&{}:{}&0.899107374019&{}:{}&0.290800702217&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.446866005564&{}:{}&0.571596675585&{}:{}&0.875269329979&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.134781121956&{}:{}&0.816224391515&{}:{}&0.318556730441&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.195090322016&{}:{}&-0.195090322016&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.980785280403&{}:{}&0.000000000000&{}:{}&1.980785280403&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.074784952616&{}:{}&0.925215047384&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)