I assume you mean bisecting (1) top-to-bottom, (2) side-to-side, (3) top-left-to-bottom-right, and (4) top-right-to-bottom-left. Actually, any straight line through the center bisects the square into two congruent (!) pieces. So there are an infinite number of ways to cut a square into two equal pieces. A very similar answer would apply if you are trying to cut the square into four equal pieces.
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it is equal because it is only a slanted square or a diamond
Yes. A rhombus is a "squashed" square in that every side is equal in length but only opposite angles are equal. The square is a special case of a rhombus where not only are opposite angles equal, but all 4 angles are equal. All squares are rhombuses but not all rhombuses are squares.
It is the square which has 4 equal sides and 4 equal interior angles
"Trisecting" is cutting into three equal parts, not four. Trisecting is impossible. Cutting into four equal parts is possible though, and trivial once you know how to bisect an angle.
Then its area will only be a quater of once it was.