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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 the given figure). Prove that ∠ACP = ∠QCD.

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Join chords AP and DQ.

For chord AP,

∠PBA = ∠ACP (Angles in the same segment) ... (1)

For chord DQ,

∠DBQ = ∠QCD (Angles in the same segment) ... (2) ABD and

PBQ are line segments intersecting at B. ∠PBA = ∠DBQ

(Vertically opposite angles) ... (3) From equations (1), (2), and

(3), we obtain ∠ACP = ∠QCD

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