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