2002 Indonesia MO Problems/Problem 4
Given a triangle with . On the circumcircle of triangle there exists point , which is the midpoint of arc that contains . Let be a point on such that is perpendicular to . Prove that .
We use the method of phantom points.
Draw and , and extend line past to a point such that . Draw point at the midpoint of , and at the intersection of the perpendicular to from and the perpendicular bisector of .
Since , we have by side-side-side similarity. Then , so is cyclic.
In particular, since we have , we know that must be the midpoint of the arc of the circumcircle of that contains point , and since was on the perpendicular to from , we must have that is the foot of the perpendicular of to . But this uniquely identifies , and we are done.
|2002 Indonesia MO (Problems)|
|1 • 2 • 3 • 4 • 5 • 6 • 7 • 8||Followed by|
|All Indonesia MO Problems and Solutions|