Segments AB and CD intersect at point O and are halved by this point. Point M is marked on segment AC

Segments AB and CD intersect at point O and are halved by this point. Point M is marked on segment AC, and point K is marked on segment BD so that AM = BK. Prove that: 1) OM = OK; 2) points M, O and K lie on one straight line

Consider triangles AOC and BOD, in them:
AO = BO (conditional)
CO = DO (conditional)
angle AOC = angle BOD (vertical angles)
It follows that the triangles AOC and BOD are equal

Next, consider the triangles AMO and BKO, in them:
AM = BK (by condition)
AO = BO (conditional)
angle MOA = angle KOB (vertical angles)
It follows that the triangles AMO and BKO are equal

In triangles that are equal to each other, the corresponding sides are equal, i.e. MO = KO

Vertical angles are formed at the intersection of two straight lines, in our case these are straight lines BA and MK, point O is the point of intersection of these straight lines, therefore, belongs to the straight line MK