A tangent to a circle is perpendicular to the radius at the point of tangency. A tangent intersects a circle at one point. Point A is the point of tangency to circle N.
As the tangent AP is perpendicular to radius AO, angle PAO is 90 degrees. Therefore any other tangent to the circle N is 90 degrees to the raidius at any points.
Example
In the figure on the right, P is a point outside the circle, with centre O, PA and PB are two tangents drawn from P to touch the circle at A and B respectively. We can find that 
i) AP = BP
ii) ÐAPO = ÐBPO
iii) ÐAOP = ÐBOP
ÐOAP = ÐOBP = 90° (tan ⊥ rad.)
△AOP and △BOP are congruent (RHS Property)
AP = BP
ÐAPO = ÐBPO and ÐAOP = ÐBOP
We can conclude that:
a) tangents drawn to a circle from an external point are equal
b) the tangents subtend equal angles at the centre
c) the line joining the external point to the centre of the circle bisects the angle between the tangents.
i) AP = BP
ii) ÐAPO = ÐBPO
iii) ÐAOP = ÐBOP
ÐOAP = ÐOBP = 90° (tan ⊥ rad.)
△AOP and △BOP are congruent (RHS Property)
AP = BP
ÐAPO = ÐBPO and ÐAOP = ÐBOP
We can conclude that:
a) tangents drawn to a circle from an external point are equal
b) the tangents subtend equal angles at the centre
c) the line joining the external point to the centre of the circle bisects the angle between the tangents.
In real life application, proffessional billard players use this theory in some of their skills. They use Tangents of the cue ball to find out the direction of where the ball will move.
Done by Ng Ding Yuan
