In triangle A B C with A B=23, A C=27, and B C=20, let D be the foot of the A altitude. Suppose \mathcal{P} is the parabola with focus A passing through B and C, and denote by T the intersection point of A D with the directrix of \mathcal{P}. Determine the value of D T^{2}-D A^{2}. (Recall that a parabola \mathcal{P} is the set of points which are equidistant from a point, called the focus of \mathcal{P}, and a line, called the directrix of \mathcal{P}.)