Mathematics
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
Join chord AP and DQ.
For chord AP,
⇒ ∠PBA = ∠ACP (Angles in the same segment are equal) …..(1)
For chord DQ,
⇒ ∠DBQ = ∠QCD (Angles in the same segment are equal) …..(2)
ABD and PBQ are line segments intersecting at B.
⇒ ∠PBA = ∠DBQ (Vertically opposite angles are equal) ….(3)
From equation (1) and (3) we get :
⇒ ∠DBQ = ∠ACP ………..(4)
From equation (2) and (4) we get :
⇒ ∠QCD = ∠ACP.
Hence, proved that ∠ACP = ∠QCD.
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