2012 USAJMO Problems/Problem 1
Given a triangle , let and be points on segments and , respectively, such that . Let and be distinct points on segment such that lies between and , , and . Prove that , , , are concyclic (in other words, these four points lie on a circle).
Since , the circumcircle of triangle is tangent to at . Similarly, since , the circumcircle of triangle is tangent to at .
For the sake of contradiction, suppose that the circumcircles of triangles and are not the same circle. Since , lies on the radical axis of both circles. However, both circles pass through and , so the radical axis of both circles is . Hence, lies on , which is a contradiction.
Therefore, the two circumcircles are the same circle. In other words, , , , and all lie on the same circle.
|2012 USAJMO (Problems • Resources)|
|First Problem||Followed by|
|1 • 2 • 3 • 4 • 5 • 6|
|All USAJMO Problems and Solutions|