Given: AO = OD and CO = OB

To prove: AC = BD

Proof :

It is given that AO = OD and CO = OB

Here line segments AB and CD are concurrent.

So,

∠AOC = ∠BOD …. As they are vertically opposite angles.

Now in ∆AOC and ∆DOB,

AO = OD,

CO = OD

Also, ∠AOC = ∠BOD

Hence, ∆AOC ≅ ∆BOD … by SAS property of congruency

So,

AC = BD … by cpct

Here,

∠ACO ≠ ∠BDO or ∠OAC ≠ ∠OBD

Hence there are no alternate angles, unless both triangles are isosceles triangle.

Hence proved that AC=BD but AC may not be parallel to BD.