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 right triangles OAP and OBP have common hypotenuse OP and equal radii OA=OB, so they are congruent by RHS. Equal tangent lengths and the bisected angle follow from corresponding parts. Example If OP=13 cm and the radius is 5 cm, then PA=PB=sqrt(132-52)=12 cm. If the angle between two tangents is 50, the line from their intersection to the centre divides it into two 25 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.

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 right triangles \(OAP\) and \(OBP\) have common hypotenuse \(OP\) and equal radii \(OA=OB\), so they are congruent by RHS. Equal tangent lengths and the bisected angle follow from corresponding parts.

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.

Tangents from an external point

Tangents from an external point