Create polyline features that radiate out from a point feature using Python [on hold]

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I’m using ArcGIS 10.1 - I have a point feature for each polygon feature (there's about 40 polygon features within a polygon feature class). I would like to write a python script that creates four line features that radiate out from the point feature and stops at the edge of each polygon feature. I ultimately want to cut the polygons into quads. How would I go about doing this? Should I find the vertices of the polygon and try to create a line with the point feature and the polygon vertices?

See image below – where the points are the center of each polygon and the purple lines are what I would like to create.





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