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Revision difference : diagonal |
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Version 3 |
| \PMlinkescapeword{adjacent} |
\PMlinkescapeword{adjacent} |
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| 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. |
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. |
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| Below is a figure showing a hexagon and all its diagonals (in red) with $X$ as one of its endpoints. |
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{center} |
\begin{center} |
| \begin{pspicture}(-8,0)(0,3) |
\begin{pspicture}(-8,0)(0,3) |
| \pspolygon(-5,0)(-3,0)(-2,1.4)(-3,3)(-5,3)(-6,1.5) |
\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](-6,1.5)(-3,0) |
| \psline[linecolor=red](-3,0)(-3,3) |
\psline[linecolor=red](-3,0)(-3,3) |
| \psline[linecolor=red](-3,0)(-5,3) |
\psline[linecolor=red](-3,0)(-5,3) |
| \rput[b](-2.7,-0.3){$X$} |
\rput[b](-2.7,-0.3){$X$} |
| \rput[l](-6,1.5){.} |
\rput[l](-6,1.5){.} |
| \rput[a](-3,3){.} |
\rput[a](-3,3){.} |
| \rput[r](-2,1.4){.} |
\rput[r](-2,1.4){.} |
| \end{pspicture} |
\end{pspicture} |
| \end{center} |
\end{center} |
| \textbf{Remarks}. |
\textbf{Remarks}. |
| \begin{itemize} |
\begin{itemize} |
| \item |
\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 $P$. |
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{center} |
\begin{center} |
| \begin{pspicture}(-8,0)(0,2) |
\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) |
\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) |
\psline[linecolor=red](-6,0.7)(-2,2) |
| \end{pspicture} |
\end{pspicture} |
| \end{center} |
\end{center} |
| \item |
\item |
| If a polygon $P$ has $n$ (distinct) vertices, then it has $\displaystyle{\frac{n(n-3)}{2}}$ diagonals. |
If a polygon $P$ has $n$ (distinct) vertices, then it has $\displaystyle{\frac{n(n-3)}{2}}$ diagonals. |
| \end{itemize} |
\end{itemize} |
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