Keep your distance

Geometry Level 5

Let \(\Delta ABC\) have integer side lengths and be right-angled at \(B\). Now suppose \(\Delta DEF\) is located within \(\Delta ABC\) with the same orientation, such that its sides are parallel to the corresponding sides of \(\Delta ABC\) and the distance between the corresponding parallel sides is \(3\).

If the area of \(\Delta ABC\) is \(4\) times that of \(\Delta DEF\) then there are \(n\) possible triangles \(\Delta ABC\) independent of orientation. If \(S\) is the sum of the lengths of the hypotenuses of these \(n\) triangles, then find \(n + S.\)

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