A Diagonal of Two Squares

Two squares, with side lengths A and B, are placed together such that the right side of square A touches the left side of square B and their bases are collinear. A diagonal is drawn from the bottom left corner of square A to the top right corner of square B.
  1. Find the area below the diagonal in square A, in terms of A and B.
  2. Find the smallest integers A < B, such that the area from part 1 is an integer. Repeat for A > B. Is there a restriction on the relationship between A and B to achieve this?
  3. Find A and B (A not equal to B) such that the diagonal is an integer regardless of which square is on the left. [I don't know if this can be solved with A and B both integers.]

Source: Original. Similar to a problem at the U. of Miss' Geometry Gambit Contest.

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