To solve the problem, we assign coordinates to the square and find the equations of the relevant lines, then calculate the intersection points and the area of the quadrilateral formed.

Step 1: Assign Coordinates

Let square (ABCD) have vertices (A(0,0)), (B(2,0)), (C(2,2)), (D(0,2)).

• (M) (midpoint of (BC)): ((2,1))
• (N) (midpoint of (CD)): ((1,2))

Step 2: Equations of Lines

• (AM): (y=\frac{1}{2}x)
• (AN): (y=2x)
• (AC): (y=x)
• (BD): (y=-x+2)

Step 3: Intersection Points

• (AM \cap BD): ((\frac{4}{3},\frac{2}{3})) (point (P))
• (AN \cap BD): ((\frac{2}{3},\frac{4}{3})) (point (Q))
• (AC \cap BD): ((1,1)) (point (O))
• (A) is the intersection of (AM), (AN), and (AC).

Step 4: Area of Quadrilateral (APOQ)

The quadrilateral (APOQ) is composed of two triangles (\triangle AOP) and (\triangle AOQ).

Area of (\triangle AOP):
Using determinant formula:
[ \text{Area} = \frac{1}{2}\left|0\left(\frac{2}{3}-1\right) + \frac{4}{3}(1-0) + 1\left(0-\frac{2}{3}\right)\right| = \frac{1}{3} ]

Area of (\triangle AOQ):
Similarly, it equals (\frac{1}{3}).

Total area: (\frac{1}{3}+\frac{1}{3}=\frac{2}{3}).

Answer: (\boxed{\dfrac{2}{3}})

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