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
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.