Find the number of diagonals that can be drawn in an n-side polygon

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The number of diagonals in an n-sided polygon can be calculated using the formula n(n-3)/2. A recursive approach is suggested, where the number of diagonals for an (n+1)-gon is expressed as D_{n+1} = D_n + n - 2. There is confusion regarding the calculation of diagonals from a single vertex, as simply multiplying by the number of vertices yields incorrect results. The discussion emphasizes the need to adjust the method to avoid overcounting. Understanding these concepts is crucial for accurately determining the number of diagonals in polygons.
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Find the number of diagonals that can be drawn in an n-side polygon.

The answer is n(n-3)/2.

I don't know how to do that.
 
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Set up a recursion for the number of diagonals in an n-gon: D_{n+1} = D_n + n - 2.
 
I don't understand how can I set up
D_{n+1} = D_n + n - 2.
 
How many diagonals can be drawn from 1 vertex? If you multiply that by the number of vertices you will get the wrong answer! Do you see why? How can you fix it?
 
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