I can think of two methods here.
Method 1. In the non-Cartesian coordinate system in which B = (0,0), C = (1,0), A = (0,1), write down the equations of the lines CD and AE. Solve the system of equations to find the coordinates of P.
Method 2. Consider the pencil of four lines consisting of AB, AE, AC, and the line through A parallel to CD. Then the cross-ratio of their respective points of intersection with BC is equal to the cross-ratio of their respective points of intersection with CD (the last of these being the point at infinity, since the fourth line is parallel to CD).