The segments AB and CD meet in their middle O. Prove that ACII BD.

We are given that the segments AB and CD meet in their middle O. Let us prove that ACII BD.

1. Consider triangles ACO and BDO: in them AO = BO, CO = OD (since the o-midpoints of the segments by condition), as well as the angles COA and BOD are equal as vertical. Thus, the triangles are equal in two sides and the angle between them.

2. Since the triangles are equal, there are equal angles opposite to equal sides, therefore, the angles of CAO and OBD are equal.

3. Consider AC, BD, AB as two lines and a secant. Since the angles by 2 points are equal, and they are crosswise, then ACII BD.