Two circles intersect at two points B and C. Through B, two line segme

1.	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» 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.

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»

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.

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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. Prove that ∠ACP = ∠QCD.

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