Symmetric axis-parallel integral hexagon

Geometry Level 4

A polygon is called a symmetric axis-parallel integral hexagon if:

  • It has six sides (a hexagon). It is also not degenerate (no vertex has an angle of \(0^\circ\) or \(180^\circ\)) and it doesn't self-intersect.
  • Its side lengths are (positive) integers.
  • Each of its sides is parallel to either the x-axis or the y-axis.
  • It has a line of symmetry: upon reflection on this line, the reflection coincides exactly with the original.

Consider a symmetric axis-parallel integral hexagon \(H\). Suppose its perimeter is \(P(H)\) units of length and its area is \(A(H)\) units of length squared (that is, \(P(H), A(H)\) are its perimeter and area respectively without the dimension). Let \(f(H) = |P(H)-A(H)|\). Determine the minimum value of \(f\).


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