Equilateral parallel hexagon in the plane

Geometry Level 3

Let hexagon \(ABCDEF\) on the cartesian plane be convex and equilateral with opposite sides parallel. Furthermore, let \(A\) be the origin, let \(B\) have coordinates \((b, 2)\), and let \(m\angle FAB = 120^\circ\). Given that the \(y\)-coordinates of the vertices of this hexagon are distinct elements from the set \(\{0, 2, 4, 6, 8, 10\}\), the area of the hexagon can be expressed as \(m \sqrt{n}\), where \(m\) and \(n\) are positive integers and \(n\) is square-free. What is \(m+n\)?

Details and Assumptions

Don't be fooled. The biconditionality of equilaterality and equiangularity only holds for triangles. In other words, equilaterality implies equiangularity (and vice versa) is only true when the figure is a triangle.

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