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AB and CD are the chords of a circle whose centre is O. They intersect each other at P. If PO be the bisector of ∠APD, prove that AB = CD.  

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Given: AB and CD are the chords of a circle whose centre is O. They interest each other at P. PO is the bisector of ∠APD. 

To Prove : AB = CD. 

Construction: Draw OR ⊥ AB and OQ ⊥ CD. 

Proof : In ∆OPR and ∆OPQ, 

∠OPR = ∠OPQ [Given] 

OP = OP [Common] 

And ∠ORP = ∠OQP [Each = 900

∴ ∆ORP ≅ ∆OPQ [By AAS] 

∴ OR = OQ [By cpctc] 

∴ AB = CD [∵Chords of a circle which are equidistant from the centre are equal] 

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