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) $PA \cdot PB = PT^2$