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Napoleon Points

\begin{figure}\begin{center}\BoxedEPSF{NapoleonPointInner.epsf scaled 750}\end{center}\end{figure}

The inner Napoleon point $N$ is the Concurrence of lines drawn between Vertices of a given Triangle $\Delta ABC$ and the opposite Vertices of the corresponding inner Napoleon Triangle $\Delta N_{AB}N_{AC}N_{BC}$. The Triangle Center Function of the inner Napoleon point is

\begin{displaymath}
\alpha=\csc(A-{\textstyle{1\over 6}}\pi).
\end{displaymath}


\begin{figure}\begin{center}\BoxedEPSF{NapoleonPointOuter.epsf scaled 750}\end{center}\end{figure}

The outer Napoleon point $N'$ is the Concurrence of lines drawn between Vertices of a given Triangle $\Delta ABC$ and the opposite Vertices of the corresponding outer Napoleon Triangle $\Delta N_{AB}'N_{AC}'N_{BC}'$. The Triangle Center Function of the point is

\begin{displaymath}
\alpha=\csc(A+{\textstyle{1\over 6}}\pi).
\end{displaymath}

See also Napoleon's Theorem, Napoleon Triangles


References

Casey, J. Analytic Geometry, 2nd ed. Dublin: Hodges, Figgis, & Co., pp. 442-444, 1893.

Kimberling, C. ``Central Points and Central Lines in the Plane of a Triangle.'' Math. Mag. 67, 163-187, 1994.




© 1996-9 Eric W. Weisstein
1999-05-25