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Let $P$ be a polygon or a polyhedron. Two vertices on $P$ are adjacent if the line segment joining them is an edge of $P$ . A 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.
\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){.} \end{pspicture}
Remarks.
- 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 $P$ .
\begin{pspicture}(-8,0)(0,2) \pspolygon(-5,0)(-4,0.5)(-2,0)(-2,2)(-3,1)(-4,1.3)(-5,1.3)(-6,2)(-6,0.7) \psline[linecolor=red](-6,0.7)(-2,2) \end{pspicture}
- If a polygon $P$ has $n$ (distinct) vertices, then it has $\displaystyle{\frac{n(n-3)}{2}}$ diagonals.
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