Segments AB and CD intersect at point O, and AO = BO, CO = DO. Prove that BC || AD.

Let’s connect points AB, BC, CD, AD. We get a quadrilateral in which the segments AB and CD are diagonals. In point O, the diagonals bisect each other. If in a quadrangle the diagonals at the intersection point are divided in half, then such a quadrangle will be a parallelogram. The opposite sides of a parallelogram are equal and parallel. Sides AD and BC are opposite sides of parallelogram ABCD, hence BC || AD.