Question

how can we prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre

Answer 1

let p be the point of contact and pt be the tangent at the point on the circlewith centre o.
since op is the radius  of the circle and pt is a tangent at p ,op perpendicular pt.
thus the perpendicular at the point of contact to the tangent passes through the centre.

Answer 2

Answer:

A tangent PR has been drawn touching thecircle at point P. Draw QP ⊥ RP at point P, such that point Q lies on thecircle. Now, above case is possible only when centre O lies on the line QP. Hence, perpendicular at the point of contact to the tangent to a circle passes through the centre of thecircle.