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Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection

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Given : Two intersecting circles, in which OO′ is the line of centres and A and B are two points of intersection. 

To prove : ∠OAO′ = ∠OBO′ 

Construction : Join AO, BO, AO′ and BO′. 

Proof : In ∆AOO′ and ∆BOO′, we have 

AO = BO [Radii of the same circle] 

AO′ = BO′ [Radii of the same circle] 

OO′ = OO′ [Common] 

∴ ∆AOO′ ≅ ∆BOO′ [SSS axiom] 

⇒ ∠OAO′ = ∠OBO′ [CPCT] 

Hence, the line of centres of two intersecting circles subtends equal angles at the two points of intersection. Proved.

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