Based on the description - connecting each corner to the opposite corner - I would do this:
1) Create your polygon
2) Convert to path
3) Split the path at two opposite nodes, resulting in two paths, one for each half of the polygon
4) Use the Extrude extension to connect the two paths, which will result in all the opposite nodes being connected
However based on the image you included, that's not actually what you want - you want every node connected to every other node. In that case I don't know of an automated way to do it, but this manual approach isn't too bad:
[list=]
[*] Create your polygon
[*] Convert to path
[*] Switch to the
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tool
[*] Set the snapping to "Snap to Nodes" and "Snap to cusp nodes"
[*] Pick a starting node, which we'll call 1. There's already a line to node 2, so the next one to connect to is node 3.
[*] Click on node 1 then double click on node 3.
[*] Click on node 1 then double click on node 4
[*] ... repeat until node 1 is connected to all the other nodes
[*] Group the whole lot
[*] Edit=>Clone=>Create Tiled Clones
[*] Hit the reset button then enter the number of nodes in your polygon (N) for "Rows" and 1 for "Columns"
[*] Set "Shift Y" in the "Per Row" column to -100% in order to make all the clones appear on top of each other
[*] Set the "Per Row" rotation to 360/N
[*] Hit the Create button
[/list]
So for a 20 sided polygon you have to draw an extra 17 lines, then create 20 clones with a rotation of 18 degrees (360/20). My attempt here took about 3 minutes - far less time than writing a description:
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Be careful of
bug #168651 though - you may be best doing this in a fresh document then copy/pasting it to the final destination.