Tangents from an external point

The radius to the point of contact is perpendicular to a tangent. If \(PA\) and \(PB\) are tangents from external point \(P\) to a circle with centre \(O\), then \(PA=PB\), and \(OP\) bisects the angle between the tangents.

The line OP is the symmetry axis of the two tangent contacts. It divides the angle between the tangents into equal parts. The two radii form right angles with their tangents; these right triangles allow Pythagoras or trigonometry to determine missing lengths.

Example

If \(OP=13\text{ cm}\) and the radius is 5 cm, then \(PA=PB=\sqrt{13^2-5^2}=12\text{ cm}\). If the angle between two tangents is \(50^\circ\), the line from their intersection to the centre divides it into two \(25^\circ\) angles.

Equal tangent lengths are measured from the same external point to the points of contact. Two unrelated tangent segments need not be equal. The diagram illustrates the relationships and is not drawn to the example dimensions.