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l is a line which intersects two concentric circle (i.e., circles with the same centre) with common centre O at A, B, C and D (as in figure). Prove that AB = CD. 

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Given: l is a line which intersects two concentric circles (i.e., circles with the same centre) with common centre O at A, B, C and D. 

To Prove: AB = CD. 

Construction: Draw OE ⊥ l 

Proof : 

∴ The perpendicular drawn from the centre of a circle to a chord bisects the chord 

∴ AE = ED .....(i) 

And BE = EC ....(ii) 

Subtracting (ii) from (i), we get 

AE - BE = ED - EC 

⇒ AB = CD

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