E, H, G, and F are midpoints of sides of the quadrilateral. Move the figure around and make a conjecture about what you see.

The quadrilateral formed by connecting the midpoints of the sides of any given quadrilateral has several interesting properties. These properties are a consequence of the Midpoint Theorem, which states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. When this theorem is applied to the four sides of a quadrilateral, the resulting quadrilateral (formed by the midpoints) has the following properties:[br][br]1. **Parallelogram Property**: The quadrilateral formed by joining the midpoints of the sides of any quadrilateral is always a parallelogram. This is because, according to the Midpoint Theorem, opposite sides of the newly formed quadrilateral are parallel.[br][br]2. **Equal Halves Property**: The parallelogram bisects the area of the original quadrilateral into two equal parts. This is a direct consequence of the fact that the diagonals of the original quadrilateral, which intersect at the center of the parallelogram, divide it into four triangles of equal area.[br][br]3. **Sides Proportionality**: Each side of the newly formed parallelogram is half the length of the diagonal over which it is drawn in the original quadrilateral. This is because the midpoint segments in a triangle are half the length of the base they are parallel to.[br][br]4. **Diagonal Property**: The diagonals of the parallelogram are congruent and bisect each other. This property emerges from the parallelogram being a special case where its diagonals not only bisect each other but are also equal in length, which is a characteristic not found in all parallelograms.[br][br]5. **Parallelism to Original Diagonals**: The sides of the parallelogram are parallel to the diagonals of the original quadrilateral. This results from how the sides of the parallelogram are formed by connecting midpoints, effectively creating lines parallel to the diagonals of the original shape.[br][br]These properties illustrate the unique relationship between a quadrilateral and the parallelogram formed by its midpoints, providing an elegant example of geometric symmetry and proportionality.