I haven't done proofs in a long time, but first I would say that you have to define all of your points, such as AB is one segment in the chord, CD is another segment in the chord, and XY crosses the chord at the points L and M.

Gather any and all theorems that you would need. You know that logicaly the line has to be perpendicular to the chord, because passing through the midpoint of the chord, means that you are making a diameter of the circle.

Maybe you could try proving that the length of AL is the same length as BL, and that CM and DL are equal, meaning that you are going to have right angles wherever XY intersects the chord.....