Let there is a circle having center O. Let AB is the tangent on the circle.

Let there is a point P on AB and another point Q inside the circle. 

Now we have to prove that the line PQ passes through O. 

Let PQ does not pass through O.

Now through O, Draw a straight line CD parallel to tangent AB. PQ intersect CD at R and also intersect AB at P.

Since CD || AB

So ∠ORP = ∠RPA

But ∠RPA = 90   (since pQ is perpendicular to AB)

So ∠ORP = 90

Now ∠ROP + ∠OPA = 180     (Co-interior angle)

=> ∠ROP + 90 = 180

=> ∠ROP = 180 - 90

=> ∠ROP = 90

So the triangle ORP has two right angles which are ∠ROP and ∠ORP. It is not possible

So our auumption is wrong.

Hence PQ passes through the point O.

Thus the perpendicular at the point of contact to the tangent to a circle always passes through the centre of the circle.