Let A B C D be a convex quadrilateral with A B=C D=10, B C=14, and A D=2 \sqrt{65}. Assume that the diagonals of A B C D intersect at point P, and that the sum of the areas of \triangle A P B and \triangle C P D equals the sum of the areas of \triangle B P C and \triangle A P D. Find the area of quadrilateral A B C D.