Radii and tangents

A tangent touches a circle at one point. It is perpendicular to the radius ending at that point of contact.

Here \(OA\perp PA\) and \(OB\perp PB\). The tangent lengths \(PA\) and \(PB\) from the same external point \(P\) are equal.

The line joining an external point \(P\) to the centre \(O\) bisects the angle between the two tangents: \(\angle APO=\angle OPB\). This requires both tangents to touch the same circle.

Why: \(OA=OB\) (radii), \(OP\) is common, and \(\angle OAP=\angle OBP=90^\circ\) (tangent perpendicular to radius). Thus \(\triangle OAP\equiv\triangle OBP\) (RHS), giving equal angles at \(P\). The proof uses the right angles and equal radii, not the conclusion it is proving.

Identify the point of contact before using a right angle. A radius to some other point on the circle is not necessarily perpendicular to the tangent.

A statement that a line looks like a tangent is not a proof. Use the stated tangent condition, or establish perpendicularity to the radius at the point on the circle.