AB, CD and XY are chords in a circle with centre O. XY cuts AB and CD in L and M, which are the midpoints of AB and CD. Prove that XY is greater than either AB or CD.
I hav no idea how to approach this problem
Look at it as an extreme value problem like this:
Since both and are chords of the same circle that intersect in the midpoint of , you have that and that .
Since , independently of the exact choice of , you'll find that the case where attains its smallest value is exactly the case where . Since we know that , we can infer from this that . Similarly for .
This is called the Chord Theorem (aka. Chord-Chord Power Theorem): it can be shown to hold by considering similar triangles, for example.
However, if what I wrote seems like so much gibberish to you, then, maybe, another approach is needed than what I have suggested. I just cannot think of a fundamentally different approach at the moment.