2015 USAJMO Problems/Problem 5
Contents
[hide]Problem
Let be a cyclic quadrilateral. Prove that there exists a point on segment such that and if and only if there exists a point on segment such that and .
Solution
Note that lines are isogonal in , so an inversion centered at with power composed with a reflection about the angle bisector of swaps the pairs and . Thus, so that is a harmonic quadrilateral. By symmetry, if exists, then . We have shown the two conditions are equivalent, whence both directions follow
Solution 2
Note that lines are isogonal in , so an inversion centered at with power composed with a reflection about the angle bisector of swaps the pairs and . Thus, so that is a harmonic quadrilateral. By symmetry, if exists, then . We have shown the two conditions are equivalent, whence both directions follow
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See Also
2015 USAJMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |