2018 USAJMO Problems/Problem 3
Contents
Problem
() Let be a quadrilateral inscribed in circle with . Let and be the reflections of over lines and , respectively, and let be the intersection of lines and . Suppose that the circumcircle of meets at and , and the circumcircle of meets at and . Show that .
Solution 1
First we have that by the definition of a reflection. Let and Since is isosceles we have Also, we see that using similar triangles and the property of cyclic quadrilaterals. Similarly, Now, from we know that is the circumcenter of Using the properties of the circumcenter and some elementary angle chasing, we find that
Now, we claim that is the intersection of ray and the circumcircle of To prove this, we just need to show that is cyclic by this definition of We have that We also have from before that so and this proves the claim.
We can use a similar proof to show that are collinear.
Now, is the radical axis of the circumcircles of and Since lies on and lie on the circumcircle of and lie on the circumcircle of we have that However, so Since are collinear and so are we can add these equations to get which completes the proof.
~nukelauncher (Monday G. Fern)
Solution 2
We begin with the following claims.
Claim. is the circumcenter of . Proof. By reflection .
Claim. is the circumcenter of . Proof. First, we have
Then
Then . This is enough to imply what we desire. \newline
Claim. is the circumcenter of . Proof. Similar to above.
Claim. are collinear. Proof. We have
Claim. are collinear. Proof. Similar to above.
Since is cyclic, . However, , so . Similarly, . Finishing, we have , as desired. ~MSC
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See also
2018 USAJMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |