Prove that the length of the tangents drawn from an external point to a circle are equal.

Given: Two tangents drawn from an external point to a circle.

To do: To Prove that the lengths of the tangents drawn from an external point to a circle are equal.

Consider the following diagram.

Let P be an external point and PA and PB be tangents to the circle.

We need to prove that PA$\perp $PB

Now consider the triangles 

$\vartriangle OAP$ and $\vartriangle OBP$

$\angle A = \angle B = 90^{o}$

$OP = OP$                                                                   [common]

$OA = OB =$ radius of the circle

Thus, by Right Angle‐Hypotenuse‐Side criterion of congruence we have,

$\vartriangle OAP\cong \vartriangle OBP$

The corresponding parts of the congruent triangles are congruent.

Thus,

PA = PB