Equal segments AB and CD intersect at point O, with AC parallel to BD. Prove that triangle AOC is isosceles.

Given:
segments AB = CD;
AB and CD meet at point O;
AC parallel to BD.
Prove that triangle AOC is isosceles.
Proof:
1) Consider triangles AOC and BOD. They have an angle ACO = angle BDO, since these are criss-crossing angles for parallel lines AC and BD, and a secant CD;
2) Angle CAO = angle DBO, since these are cross-lying angles for parallel lines AC and BD, and secant AB;
3) Angle AOC = angle DOB, as they are vertical.
Therefore, the triangles AOC = BOD, then AO = OB = CO = OD, since CD = AB.
Then triangle AOC is isosceles.