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

Answer 1

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