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Dividing the altitude of a triangle into four equal parts

Source: RMO 2014 West Bengal Problem 1

December 7, 2014
geometrytrapezoidsimilar trianglesgeometry proposed

Problem Statement

In acute △ABC,\triangle ABC, let DD be the foot of perpendicular from AA on BCBC. Consider points K,L,MK, L, M on segment ADAD such that AK=KL=LM=MDAK= KL= LM= MD. Suppose the sum of the areas of the shaded region equals the sum of the areas of the unshaded regions in the following picture. Prove that BD=DCBD= DC.
http://s27.postimg.org/a0d0plr4z/Untitled.png
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