diagonal (original) (raw)

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.

Remarks.

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  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}(-227.62204pt,0.0pt)(0.0pt,56.905502pt)\leavevmode% \ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces% \ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces% \ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces% \special{pst: \pst@dict\tx@STP\pst@newpath\psk@origin\psk@swapaxes\pst@code end }\ignorespaces\leavevmode\ignorespaces\ignorespaces\ignorespaces\ignorespaces% \special{pst: \pst@dict\tx@STP\pst@newpath\psk@origin\psk@swapaxes\pst@code end }\ignorespaces\end{pspicture}
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  If a polygon P has n (distinct) vertices, then it has n⁢(n-3)2 diagonals.