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. Prove that ∠ACP = ∠QCD.

Answer 1

Two circles are drawn taking PQ and PR of a triangle as diameter. 

Let these intersect at P and S. 

To Prove: ∠ACP = ∠QCD 

Proof: ∠ABP = ∠QBD ………….. (i) 

(vertically opposite angles) 

∠ABP = ∠ACP ……….. (ii) 

(angles in the same segment) 

Similarly, ∠QCD = ∠QBD …………. (iii)(angles in the same segment) 

From (i), (ii), and (iii), 

∠ACP = ∠QCD.