Heptagon Theorem

Author

Eric W. Weisstein
May 3, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/HeptagonTheorem.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HeptagonTheorem.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Stuff

Let $H$ be a heptagon with seven vertices given in cyclic order inscribed in a \hrefacc{conic}{Conic Section}. Then the
\href{Pascal lines} of the seven \hrefacc{hexagons}{Hexagon} obtained by omitting each vertex of $H$ in turn and keeping
the remaining vertices in the same cyclic order are the sides of a \href{heptagon} $I$ which circumscribes a
\hrefacc{conic}{Conic Section}.

t=Mod[Table[2Pi/7 i+Random[Real,{-1,1}.2],{i,1,14,2}],2Pi]

p={Cos[#],1.2Sin[#]}&/@t

t6=Table[Drop[t,{i}],{i,7}]

pts=Table[PascalPoints[Drop[p,{i}]],{i,7}]

ptsu=Union[Flatten[pts,1]]

Show[Graphics[{
ParametricPlot[{Cos[t],1.2Sin[t]},{t,0,2Pi},DisplayFunction->Identity,Axes->False][[1]],
RGBColor[0,0,1],PointSize[.03],Point/@p,line[p],
RGBColor[1,0,0],PascalLines[1.2,#,10][[1]]&/@t6,
Map[Point,pts,{2}]
},AspectRatio->Automatic,PlotRange->2{{-1,1},{-1,1}}]]