2017 USAJMO Problems/Problem 3
() Let be an equilateral triangle and let be a point on its circumcircle. Let lines and intersect at ; let lines and intersect at ; and let lines and intersect at . Prove that the area of triangle is twice that of triangle .
WLOG, let . Let , and . After some angle chasing, we find that and . Therefore, ~ .
Lemma 1: If , then . This lemma results directly from the fact that ~ ; , or .
Lemma 2: . We see that , as desired.
Lemma 3: . We see that However, after some angle chasing and by the Law of Sines in , we have , or , which implies the result.
By the area lemma, we have and .
We see that . Thus, it suffices to show that , or . Rearranging, we find this to be equivalent to , which is Lemma 3, so the result has been proven.
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