Problem 534: Right Triangle, Squares, Perpendicular, Dissection, Areas. Level: High School, SAT Prep, College Geometry

The figure shows a right triangle ABC (BC = a, AB = c) with the squares ABDE and ACFG. Line BHJ is perpendicular to BE. Line CK is perpendicular to BC. Lines BE and GA extended meet at M. Lines BJ and EA extended meet at L. Lines LN and KP are perpendicular to BJ. If A₁, A₂, A₃, A₄, A₅, A₆, A₇, A₈, and A₉ are the shaded areas, prove that (1) , (2) , (3) Area ABDE = Area AHLN + Area FJKP.