Difference between revisions of "1990 IMO Problems/Problem 1"
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==Solution 1== | ==Solution 1== | ||
− | With simple angle chasing, we find that triangles <math>CEG</math> and <math> | + | With simple angle chasing, we find that triangles <math>CEG</math> and <math>BMD</math> are similar. |
− | so, <math> | + | so, <math>/frac{MB}{EC} = /frac{MD}{EG}</math>. (*) |
− | + | Again with simple angle chasing, we find that triangles <math>CEF</math> and <math>AMD</math> are similar. | |
− | so, <math> | + | so, <math>/frac{MA}{EC} = /frac{MD}{EF}</math>. (**) |
− | so, by (*) and (**), we have <math> | + | so, by (*) and (**), we have <math>/frac{EG}{EF} = /frac{MA}{MB} = /frac{t}{1-t}</math>. |
This solution was posted and copyrighted by e.lopes. The original thread for this problem can be found here: [https://aops.com/community/p366701] | This solution was posted and copyrighted by e.lopes. The original thread for this problem can be found here: [https://aops.com/community/p366701] |
Revision as of 06:16, 17 November 2024
Contents
[hide]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 |