Question 14
Let $\overline{AB}$ be a diameter of circle $\omega.$ Extend $\overline{AB}$ through $A$ to $C.$ Point $T$ lies on $\omega$ so that line $CT$ is tangent to $\omega.$ Point $P$ is the foot of the perpendicular from $A$ to line $CT.$ Suppose $AB = 18,$ and let $m$ denote the maximum possible length of segment $BP.$ Find $m^2.$