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Let $A B C$ be a triangle and let $A D$ be the perpendicular from $A$ on to $B C$. Let $K, L, M$ be points on $A D$ such that $A K=K L=$ $L M=M D$. If the sum of the areas of the shaded regions is equal to the sum of the areas of the unshaded regions, prove that $B D=D C$.

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