2018 USAMO Problems/Problem 5
In convex cyclic quadrilateral we know that lines and intersect at lines and intersect at and lines and intersect at Suppose that the circumcircle of intersects line at and , and the circumcircle of intersects line at and , where and are collinear in that order. Prove that if lines and intersect at , then
so are collinear. Furthermore, note that is cyclic because: Notice that since is the intersection of and , it is the Miquel point of .
Now define as the intersection of and . From Pappus's theorem on that are collinear. It’s a well known property of Miquel points that , so it follows that , as desired. ~AopsUser101