This example came up in my college geometry class. It's a circle with a secant and a tangent that intersect in the exterior of the circle. (Originally, the radius of the circle was unknown). My professor solved it by drawing an auxiliary line from A to D and then using similar triangles, but I realized that the Secant-Tangent Theorem should also apply here--but it doesn't appear to work.
I've recreated the situation in Geogebra and measured all of the relevant segments. The tangent FD = 12, while the secant FE is comprised of two segments, FG = 6 (external) and GE = 18 (internal). But by this Secant-Tangent Theorem, this would mean that: $12^2=(6)(18)$ which is a contradiction. Is there something I'm missing, or is there a reason that the Secant-Tangent Theorem doesn't apply here?
The issue here is in your application of the secant tangent theorem. For the secant line, you need to use the entire length of the secant from the intersection point all the way to the other side of the circle. Thus, as suggested by @wilhelm_iv, we should have $12^2 = 6 * (18+6)$.