Consider a circle and consider two tangents touching the circle that meet at a some point \(P\). Let point \(O\) be at the center of the circle and let \(A\) and \(B\) be the points where the two tangents touch the circle:
The lines \(\overline{OA}\) and \(\overline{OB}\) are both radii, and therefore equal in length:
Consider the line \(\overline{OP}\). It's a line shared by the triangles, \(PAO\) and \(PBO\):
Since the angle between a radius and a tangent is always \(90^\circ\), then \(\angle OAP = \angle OBP = 90^\circ\). This means triangles \(PAO\) and \(PBO\) are both right-angled. Since two sides of \(PAO\) are the same length as two sides of \(PBO\), then the third side should also be equal:
This means the tangents \(\overline{PA}\) and \(\overline{PB}\) have the same length.