If each diagonal of a quadrilateral divides it into two triangles of equal areas, then prove that the quadrilateral is a parallelogram.

Given: ABCD is a quadrilateral where each diagonal divides it into two triangles of equal areas.

To prove: ABCD is a parallelogram.

Proof: Since each diagonal divides the quadrilateral into two triangles of equal areas, we have:

Area of Δ ABC = $\dfrac{1}{2}$ Area of ABCD ……………….(1)

Area of Δ ABD = $\dfrac{1}{2}$ Area of ABCD ……………….(2)

Area of Δ BCD = $\dfrac{1}{2}$ Area of ABCD ……………….(3)

From equations (1) and (2), we get:

Area of Δ ABC = Area of Δ ABD

Since Δ ABC and Δ ABD lie on same base AB and have equal areas, they must lie between the same parallel lines.

AB ∥ CD

Similarly, from equations (1) and (3), we get:

Area of Δ ABC = Area of Δ BCD

Since Δ ABC and Δ BCD lie on same base BC and have equal areas, they must lie between the same parallel lines.

BC ∥ AD

Since opposite sides are parallel, ABCD is a parallelogram.

Hence, ABCD is a parallelogram.