Proof that a tangent is

perpendicular to the

radius drawn to the point of tangency:

Givens are

circle O, line AB tangent to O at point A, and radius OA.

We'll use an

indirect proof; that is, we'll begin by assuming that OA is not perpendicular to AB. Therefore, there must be some

line segment that

*is* perpendicular to AB; we'll call it OX (where X is a point on AB). A perpendicular line is the shortest distance to a point from a line; therefore, OX is the shortest distance to O from AB, and is shorter than OA. But a radius is always shorter than a line segment from the

center of a circle to a point in the

exterior of the circle, so OA is shorter than OX. We have reached a

contradiction, and have proved the initial assumption false.

Q.E.D.