tangent-radius perpendicularity

Roll a ball against a wall and the point where it just kisses the wall is special: the line straight out from the ball's centre to that point hits the wall square-on, at a right angle. The same is true of a circle and its tangent line: where a tangent touches a circle, the radius to that point of contact is perpendicular to the tangent.

Precisely: if a line is tangent to a circle at a point T, then the radius OT (O the centre) is perpendicular to the tangent line at T. The reason is that the tangent meets the circle only at T, so T is the single closest point of the line to O, and the shortest segment from a point to a line is the perpendicular one — hence OT _|_ tangent. The converse holds too: a line through a point T of the circle that is perpendicular to the radius OT must be the tangent there. A powerful companion result, the two-tangents (equal tangents) theorem, follows: from an external point P outside the circle, the two tangent segments drawn to the circle have equal length, PT_1 = PT_2, because the two right triangles OT_1P and OT_2P share hypotenuse OP and have equal radius-legs, so they are congruent.

These facts are everywhere in circle constructions and proofs: they let you drop a tangent from an outside point, prove that a quadrilateral has an inscribed circle exactly when its opposite sides have equal sums (Pitot's theorem, via equal tangents), and they set up the tangent-chord angle and tangent-secant power relations. The misconception to head off: the perpendicularity is between the tangent and the radius drawn TO THE POINT OF CONTACT — not to some other point of the circle. And the tangent touches at exactly one point; a line through that point in any other direction would be a secant, cutting the circle twice.