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.