Consider a circle and two tangents that meet at point \(P\). Let point \(A\) and point \(B\) be the points were the two tangents touch the circle. Also, let point \(O\) be at the center:
The lines \(\overline{OA}\) and the lines \(\overline{OB}\) 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 in \(90^\circ\), then \(\angle OAP = \angle OBP = 90^\circ\). This means triangles \(PAO\) and \(PBO\) are both right-angled. Since two sides of \(PAO\) is the same length as two of the same sides of \(PBO\), then the third side should also be equal:
This means the tangents \(\overline{PA}\) and \(\overline{PB}\) have the same length.