Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle.

Draw a circle with centre O.
draw a tangent PR touching circle at P.
Draw QP perpendicular to RP at point P, Qp lies in the circle.
Now, angle OPR = 90 degree (radius perpendicular to tangent)
also angle QPR = 90 degree (given)
 

Therefore angle OPR = angle QPR. This is possible only when O lies on QP. Hence, it is proved that the perpendicular at the point of contact to the tangent to a circle passes through the centre. 
 

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 Draw a cricle with centre o

draw a tangent pr touching circle at p

draw qp perpendecular to rp at p, qp lies in in the circle

now, angle QPR=90 degree(radius perpendecular to tangent)

also angleQPR =90degree  (given)

therefore angle oqr =angle qpr possible only when o lies on qp ,proved