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area chasing, square, rhombus, symmetric (2018 Romanian NMO VII P2)

Source:

June 3, 2020
geometryrhombusareasquaresymmetry

Problem Statement

In the square ABCDABCD the point EE is located on the side [AB][AB], and FF is the foot of the perpendicular from BB on the line DEDE. The point LL belongs to the line DEDE, such that FF is between EE and LL, and FL=BFFL = BF. NN and PP are symmetric of the points A,FA , F with respect to the lines DE,BLDE, BL, respectively. Prove that:
a) The quadrilateral BFLPBFLP is square and the quadrilateral ALNDALND is rhombus. b) The area of the rhombus ALNDALND is equal to the difference between the areas of the squares ABCDABCD and BFLPBFLP.
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