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

Let we have a circle with center O.

At point P, a tangent PR has been drawn touching the circle.

From point Q lies on the circle, draw QP ⊥ RP at point P.

∠OPR = 90° (QP ⊥ RP)

Also, ∠QPR = 90° (Given)

Since ∠OPR = ∠QPR (each 90°), it 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 the circle.