GeoProof — Tangent ⟂ Radius explorer

the tangent at T is perpendicular to the radius OT

Tangent ⟂ Radius

A tangent is a line that touches a circle at exactly one point. At that point of contact, the tangent is always perpendicular to the radius drawn there:

tangent at T ⟂ radius OT (a 90° angle)

Drag T around the circle and the right angle between the radius and the tangent never changes.

Why it's true

The radius OT is the shortest distance from the centre O to the tangent line. The shortest segment from a point to a line is always the perpendicular one — so OT must meet the tangent at 90°. Turn on "Nearest-point check": slide Q along the tangent and OQ is smallest exactly at T, where it equals the radius.

Only one tangent per point

Because the direction is fixed (perpendicular to the radius), there is exactly one tangent line at each point of the circle.

Where it's used

This fact is the starting point for constructing tangents, for the tangent–chord angle, for equal tangent lengths from an external point, and for tangent circles and many engineering and optics problems.