In the figure, P is a point on the chord BC such that AB = AP. Prove that, CP = CQ.
Given P is a point on the chord BC and AB = AP
... ∠ABP = ∠APB ...........(i)
∠ABP = ∠AQC ..............(ii)[angle in the same segment of a circle] and
∠APB = ∠CPQ...............(iii) [Vertically opposite angles]
... ∠CPQ = ∠ AQC = ∠PQC [from (i)]
⇒ CQ = PC
or CP = CQ
... ∠ABP = ∠APB ...........(i)
∠ABP = ∠AQC ..............(ii)[angle in the same segment of a circle] and
∠APB = ∠CPQ...............(iii) [Vertically opposite angles]
... ∠CPQ = ∠ AQC = ∠PQC [from (i)]
⇒ CQ = PC
or CP = CQ
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