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Sum of Areas is Equal

Source: 2018 AIME II #12

March 23, 2018
AIMEAIME IIgeometry

Problem Statement

Let ABCDABCD be a convex quadrilateral with AB=CD=10AB=CD=10, BC=14BC=14, and AD=265AD=2\sqrt{65}. Assume that the diagonals of ABCDABCD intersect at point PP, and that the sum of the areas of △APB\triangle APB and △CPD\triangle CPD equals the sum of the areas of △BPC\triangle BPC and △APD\triangle APD. Find the area of quadrilateral ABCDABCD.
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