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Let $$l_1$$, $$l_2$$ be any two parallel lines and $$B$$, $$C$$ be any two points on $$l_1$$ and $$A_1, A_2, \ldots, A_{2010}$$ be points on $$l_2$$. If $$\Delta_i$$ denotes the area of the triangle $$A_i BC$$ and if $$\sum_{i=1}^{2010} \Delta_i = 2010$$, Then the area of $$\triangle A_{2010}BC$$ is
All the triangles $$A_i BC$$ have the same base $$BC$$ and the same height, the distance between the parallel lines. So all 2010 areas are equal, and since they add up to 2010, each is 1. The area of $$\triangle A_{2010}BC$$ is 1.
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