qgis - Transforming Visio polygon to shapefile?

I am a complete newb at this stuff and don't know what I don't know. I may use incorrect language.

Essentially, I want to convert the shapes in a Microsoft Visio diagram into a shapefile. I pretty much want the diagram to BECOME a shapefile.

I have a csv file which describes 10-15 arbitrary polygons from that Visio diagram:

ShapeNo ShapeName PointNo   X    Y
1       My Square   2       37  155
1       My Square   4       116 155
1       My Square   6       116 234
1       My Square   8       37  234

1       My Square   10      37  155

Polygons are going to be squares, rectangles, with the occasional circle/oval. I'll generate the file above with a script which exports each point of each shape in the diagram. Part and parcel of the X/Y info is the relation of each shape to the others.

I'm hoping QGIS can take this information and essentially create a shapefile for me based on some arbitrary lat/lon "center".

Can this be done? If so, a push in the right direction (articles, keywords, walk-throughs) would be really appreciated.