Tangents to a Circle

A tangent is a straight line that just grazes a circle, touching it at exactly one point — the point of contact. Where it touches, the tangent makes a perfect right angle with the radius drawn to that point.

\text{tangent} \perp \text{radius at the point of contact}

Draw two tangents from the same outside point and something neat happens: they have equal length, so the two tangents, the two radii and the centre fold into a symmetric kite.

For a line touching a circle:

a tangent touches the circle at exactly one point;
the tangent is perpendicular to the radius at the point of contact — the angle there is 90^\circ;
the two tangents from one external point are equal in length;
so the external point, the two points of contact and the centre form a symmetric kite, with the two right angles at the contact points.

Why it works

Watch a single external point send two tangents to the circle. Step through it and notice the two right angles and the two equal lengths.

Because both right-angled triangles OT_1E and OT_2E share the side OE and have equal radii, they are mirror images — so the tangent lengths ET_1 and ET_2 match.