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

OP is the radius of circle. O′P is perpendicular to the tangent at the point of contact and it doesn't pass through the center ∠O′PT = 90°. (1) From the property, the radius of the circle is perpendicular to the tangent at the point of contact. ∠OPT = 90°. (2) (1) and (2) are contradicting each other. Both (1) and (2) can be true only if O′ lies on point O. Therefore, O′P at the point of contact to the tangent passes through the center.