Two tangent TP and TQ are …

Given A circle with centre O and an external point T and two tangents TP and TQ to the circle, where P, Q are the points of contact.
To Prove: {tex}\angle{/tex}PTQ = 2{tex}\angle{/tex}OPQ
Proof: Let {tex}\angle{/tex}PTQ = {tex}\theta{/tex}
Since TP, TQ are tangents drawn from point T to the circle.
TP = TQ
{tex}\therefore{/tex} TPQ is an isoscles triangle
{tex}\therefore{/tex} {tex}\angle{/tex}TPQ = {tex}\angle{/tex}TQP = {tex}\frac12{/tex} (180^o - {tex}\theta{/tex}) = 90^o - {tex}\fracθ2{/tex}
Since, TP is a tangent to the circle at point of contact P
{tex}\therefore{/tex} {tex}\angle{/tex}OPT = 90^o
{tex}\therefore{/tex} {tex}\angle{/tex}OPQ = {tex}\angle{/tex}OPT - {tex}\angle{/tex}TPQ = 90^o - (90^o - {tex}\frac12{/tex} {tex}\theta{/tex}) = {tex}\fracθ2{/tex}= {tex}\frac12{/tex}{tex}\angle{/tex}PTQ
Thus, {tex}\angle{/tex}PTQ = 2{tex}\angle{/tex}OPQ