Let P be a point exterior to a circle C of radius r and PT be a line

tangent to circle C at T. Also let d be the distance from P to the

center of C. Then (PT)² = d² - r² = P(C).

Let O be the center of circle C.

If we construct a radius OT, then by the Perpendicular at Tangent Point theorem,

it is perpendicular to the tangent line PT.

Therefore angle PTO is a right angle.



Since we have a right triangle PTO, we can use the Pythagorean Theorem to state

that:

d² = r² + (PT)².
Therefore:
(PT)² = d² - r² ,
which by definition is P(C).

Q.E.D.