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sum of ratios

Source: flanders '03

September 28, 2005
ratiogeometry

Problem Statement

Through an internal point OO of ΔABC\Delta ABC one draws 3 lines, parallel to each of the sides, intersecting in the points shown on the picture. http://www.mathlinks.ro/Forum/album_pic.php?pic_id=289 Find the value of ∣AF∣∣AB∣+∣BE∣∣BC∣+∣CN∣∣CA∣\frac{|AF|}{|AB|}+\frac{|BE|}{|BC|}+\frac{|CN|}{|CA|}.
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