diagonalDiagonal2007-10-12 10:12:162007-10-29 17:41:22Definitionadjacent vertices\usepackage{amssymb,amscd}
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\newcommand{\pdiff}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\mpdiff}[3]{\frac{\partial^#1 #2}{\partial #3^#1}}\PMlinkescapeword{adjacent}
Let $P$ be a polygon or a polyhedron. Two vertices on $P$ are \emph{adjacent} if the line segment joining them is an edge of $P$. A \emph{diagonal} of $P$ is a line segment joining two non-adjacent vertices.
Below is a figure showing a hexagon and all its diagonals (in red) with $X$ as one of its endpoints.
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\begin{pspicture}(-8,0)(0,3)
\pspolygon(-5,0)(-3,0)(-2,1.4)(-3,3)(-5,3)(-6,1.5)
\psline[linecolor=red](-6,1.5)(-3,0)
\psline[linecolor=red](-3,0)(-3,3)
\psline[linecolor=red](-3,0)(-5,3)
\rput[b](-2.7,-0.3){$X$}
\rput[l](-6,1.5){.}
\rput[a](-3,3){.}
\rput[r](-2,1.4){.}
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\textbf{Remarks}.
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\item
If $P$ is convex, then the relative interior of a diagonal lies in the relative interior of $P$. Below is a figure showing that a diagonal may partially lie outside of $P$.
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\psline[linecolor=red](-6,0.7)(-2,2)
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\item
If a polygon $P$ has $n$ (distinct) vertices, then it has $\displaystyle{\frac{n(n-3)}{2}}$ diagonals.
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