2010 IMO Problems/Problem 2
Problem
Given a triangle , with as its incenter and as its circumcircle, intersects again at . Let be a point on arc , and a point on the segment , such that . If is the midpoint of , prove that the intersection of lines and lies on .
Authors: Tai Wai Ming and Wang Chongli, Hong Kong
Solution
Note that it suffices to prove alternatively that if meets the circle again at and meets at , then is the midpoint of . Let meet at .
Observation 1. D is the midpoint of arc because it lies on angle bisector .
Observation 2. bisects as well.
Key Lemma. Triangles and are similar. Proof. Because triangles and are similar by AA Similarity (for and both intercept equally sized arcs), we have . But we know that triangle is isosceles (hint: prove ), and so . Hence, by SAS Similarity, triangles and are similar, as desired.
Observation 3. As a result, we have .
Observation 4. .
Observation 5. If and intersect at , then is cyclic.
Observation 6. Because , we have .
Observation 7. is a parallelogram, so its diagonals bisect each other, so is the midpoint of , as desired.
See Also
2010 IMO (Problems) • Resources) | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |