My instructor for complex analysis used the following geometric fact without giving proof, and I'm wondering if anyone out there could help me see a good proof of this:

Let A, B, C be collinear, and suppose B and C lie on a circle with center O. Draw a line from A to the circle so that it is tangent to the circle at a point P on the circle. Prove that

$\displaystyle |AP|^2=|AB||AC|$

I'd appreciate any solutions or hints. I am hoping for a more-or-less "pure geometry" proof, rather than some kind of coordinate-based or vector-based proof, but if you find this inconvenient, I'd be happy with any solution.