Question 12
Let $ABCD$ be a convex quadrilateral with $AB = CD = 10,$ $BC = 14,$ and $AD = 2\sqrt{65}.$ Assume that the diagonals of $ABCD$ intersect at point $P,$ and that the sum of the areas of triangles $APB$ and $CPD$ equals the sum of the areas of triangles $BPC$ and $APD.$ Find the area of quadrilateral $ABCD.$