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I have this purple polygon illustrated in the figure below. I would like to split the polygon so that each line segment (green) split the polygon evenly with the surrounding line segments. A line segment is split at intersection points.

I have tried to illustrate by painting a red line where I think the polygon should be divided.

Is this a normal and simple process to do or does this require a lot of working?

The terms Voronoi and Delaunay comes to my mind, but I'm not sure if that is what I want here.

Edit:
I'll try to describe how I think of evenly in this case. I want each green line segment to take its share of the purple polygon. It is equal to buffering the green line segments, but instead of creating one big polygon (just like the purple one) each buffered line segment continue to grow until it meet another one - just like inflating a lot of ballons in the same room, they can grow as long as there are vacant space.

enter image description here

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What do you mean by "split the polygon evenly"? There is no evidence in your example that anything is "even," either in the sense of equal length, equal perimeter, or equal area. Are you perhaps asking to make a Euclidean allocation of the polygon according to the arcs determined by the intersection points? –  whuber Jul 12 '13 at 15:01
    
Throw light on "I would like to split the polygon so that each line segment (green) split the polygon evenly with the surrounding line segments." –  Naresh Jul 12 '13 at 15:07
    
whuber: Euclidean allocation could be what I'm looking for, but I'm not completely sure. I have edited my question, does it explain what I want to do now? –  Chau Jul 15 '13 at 12:53

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