Given: AO = OB , DO = OC

To prove: AC=BD and AC||BD

Proof:

It is given that, O is the midpoint of each of the line segments AB and CD.

This implies that AO = OB and DO = OC

Here line segments AB and CD are concurrent.

So,

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

Now in ∆AOC and ∆BOD,

AO = OB,

OC = OD

Also, ∠AOC = ∠BOD

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

So,

AC = BD … by cpct

∴ ∠ACO = ∠BDO … by cpct

But ∠ACO and ∠BDO are alternate angles.

∴ We conclude that AC is parallel to BD.

Hence we proved that AC=BD and AC||BD