2020 CIME I Problems/Problem 9
Problem 9
Let be a cyclic quadrilateral with . Let be the point on such that . Then can be expressed in the form , where and are relatively prime positive integers. Find .
Solution
Let be the reflection of over line . Since , are collinear. Suppose and are the projections of and onto line , respectively. We want to find which by similar triangles is also equal to from . Since , this also equals . We know that and each share the same base, so this can also be interpreted as . The sine area formula gives Quadrilateral is cyclic, so because both angles subtend arc on the circumcircle of Quadrilateral . We can then replace every with , but realise that if we do that, the s will cancel out. The requested area ratio is thus . The answer is .
Solution 2 (Law of Sines)
We look for the ratio so thus we use the Law of Sines since it involves ratios.
By the Law of Sines used on and ,
Since implies , this implies
Now we just need to find or its reciprocal to get the answer.
We use Law of Sines again on and as follows:
Hence .
Thus .
The answer is .
~FIREDRAGONMATH16
Video Solution
https://www.youtube.com/watch?v=atUCE3oSieg&lc=UgwRISSUhBk6GBF9g294AaABAg
See also
2020 CIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All CIME Problems and Solutions |
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