1994 USAMO Problems/Problem 3
Problem
A convex hexagon is inscribed in a circle such that and diagonals , and are concurrent. Let be the intersection of and . Prove that .
Solution
Let the diagonals , , meet at .
First, let's show that the triangles and are similar.
because ,, and all lie on the circle, and . because , and ,,, and all lie on the circle. Then,
Therefore, and are similar, so .
Next, let's show that and are similar.
because ,, and all lie on the circle, and . because ,, and all lie on the circle. because , and ,,, and all lie on the circle. Then,
Therefore, and are similar, so .
Lastly, let's show that and are similar.
Because and ,, and all lie on the circle, is parallel to . So, and are similar, and .
Putting it all together, .
Borrowed from https://mks.mff.cuni.cz/kalva/usa/usoln/usol943.html
See Also
1994 USAMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.