A and B are the points on ⨀(O, r). is not a diameter of the circle. Prove that the tangents to the circle at A and B are not parallel.

Given that A and B are points on (O, r) and AB is a not a diameter of the circle.

We have to prove that the tangents to the circle at A and B are not parallel.

Proof:

Using the method of contradiction,

Let l and m be two parallel tangents to the circle have centre O drawn at the points A and B.

∴ OA ⊥ l and OB ⊥ m

Consider OA and OB perpendicular to l and m respectively and O is a common point.

∴ since l and m are two parallel lines, A – O – B

Hence, AB is a diameter, which contradicts with our assumption.

∴ Our assumption is wrong i.e. l and m are intersecting lines.

∴ Tangents to the circle at A and B are not parallel.

Hence proved.