Question

Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see Fig.). Prove that ∠ ACP = ∠QCD.

Answer 1

Given : Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively. 

To Prove : ∠ACP = ∠QCD. 

Proof : ∠ACP = ∠ABP ...(i) [Angles in the same segment] 

∠QCD = ∠QBD ..(ii) [Angles in the same segment] 

But, ∠ABP = ∠QBD ..(iii) [Vertically opposite angles] 

By (i), (ii) and (ii) we get 

∠ACP = ∠QCD Proved