2019 IMO Problems/Problem 2
In triangle , point lies on side and point lies on side . Let and be points on segments and , respectively, such that is parallel to . Let be a point on line , such that lies strictly between and , and . Similarly, let be the point on line , such that lies strictly between and , and .
Prove that points , and are concyclic.
The essence of the proof is to build a circle through the points and two additional points and then we prove that the points and lie on the same circle.
We assume that the intersection point of and lies on the segment If it lies on segment then the proof is the same, but some angles will be replaced with additional ones up to .
Let the circumcircle of be . Let and be the points of intersection of and with . Let
since they intersept the arc of the circle .
is cyclic (in circle )
since they intersept the arc of the circle So is cyclic.
(they intersept the arc of the circle
(since they intersept the arc of the circle
Hence the point lies on
Similarly, point lies on