2008 AIME I Problem 14

Let \overline{A B} be a diameter of circle \omega. Extend \overline{A B} through A to C. Point T lies on \omega so that line C T is tangent to \omega. Point P is the foot of the perpendicular from A to line C T. Suppose A B=18, and let m denote the maximum possible length of segment B P. Find m^{2}.