1990 IMO Problems/Problem 1
Contents
Problem
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 .
Solution 1
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: [1]
Solution 2
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: [2]
See Also
1990 IMO (Problems) • Resources | ||
Preceded by First Question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |