2003 AIME II Problems/Problem 14
Let and be points on the coordinate plane. Let be a convex equilateral hexagon such that and the y-coordinates of its vertices are distinct elements of the set The area of the hexagon can be written in the form where and are positive integers and n is not divisible by the square of any prime. Find
The y-coordinate of must be . All other cases yield non-convex and/or degenerate hexagons, which violate the problem statement.
Letting , and knowing that , we can use rewrite using complex numbers: . We solve for and and find that and that .
The area of the hexagon can then be found as the sum of the areas of two congruent triangles ( and , with height and base ) and a parallelogram (, with height and base ).
From this image, we can see that the y-coordinate of F is 4, and from this, we can gather that the coordinates of E, D, and C, respectively, are 8, 10, and 6.
Let the angle between the -axis and segment be , as shown above. Thus, as , the angle between the -axis and segment is , so . Expanding, we have
Isolating we see that , or . Using the fact that , we have , or . Letting the side length of the hexagon be , we have . After simplification we find that that .
In particular, note that by the Pythagorean theorem, , hence . Also, if , then , hence and thus . Using similar methods (or symmetry), we determine that , , and . By the Shoelace theorem,
Hence the answer is .
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