2015 USAJMO Problems/Problem 5
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 .
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
All angles are directed*. Note that lines are isogonal in and are isogonal in . From the law of sines it follows that
Therefore, the ratio equals
Now let be a point of such that . We apply the above identities for to get that . So , the converse follows since all our steps are reversible.
- Beware that directed angles, or angles , are not standard olympiad material. If you use them, make sure to define and indicate what they mean. [Mentioned in Evan Chen's Geometry Olympiad book.]
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