Question

O is the centre of a circle in which chords AB and CD intersect at P such that PO bisects angle BPD . Prove that AB = CD.

Answer 1

Given: 
AB and CD are the chords of a circle with center O.
They intersect at P.
PO is bisector of ∠APD

Prove: AB = CD

Construction: 
Draw OR ⊥ AB
OQ ⊥ CD

Proof: 

In ΔOPR and ΔOPQ

∠OPR = ∠OPQ ....given

OP = OP .....Common in both

∠ORP = ∠OQP ....Construction

ΔOPR ≅ ΔOPQ ......AAS axiom

∴ OR = OQ (C.P.C.T)

AB = CD.....chords of a circle\

Hence proved.