24. Exemplar example problems V

In given figure, from an external point \(P\), a tangent \(PT\) and a line segment \(PAB\) is drawn to a circle with centre \(O\). \(ON\) is perpendicular on the chord \(AB\). Prove that:

 

(i) \(PA \cdot PB = PN^2 - AN^2\)

 

(ii) \(PN^2 - AN^2 = OP^2 - OT^2\)

 

(iii) \(PA \cdot PB = PT^2\)