Difference between revisions of "1990 IMO Problems/Problem 1"
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With simple angle chasing, we find that triangles <math>CEG</math> and <math>BMD</math> are similar. | 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. | 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 07:17, 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 | ||
