1990 IMO Problems/Problem 1
Chords and of a circle intersect at a point inside the circle. Let be an interior point of the segment . The tangent line at to the circle through , and intersects the lines and at and , respectively. If , find in terms of .
With simple angle chasing, we find that triangles and are similar.
so, . (*)
again with simple angle chasing, we find that triangles and are similar.
so, . (**)
so, by (*) and (**), we have .
This solution was posted and copyrighted by e.lopes. The original thread for this problem can be found here: 
This problem can be bashed with PoP and Ratio Lemma. Rewriting the given ratio gets . By Ratio Lemma, . Similarly, . We can rewrite these equalities to get and . Using Ratio Lemma, and . Since , we have (eq 1). Note that by Ratio Lemma, . Plugging this into (eq 1), we get . So .
This solution was posted and copyrighted by AIME12345. The original thread for this problem can be found here: 
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