Finding the midpoint of a Quadrilateral

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Given any quadrilateral, construct a midpoint on each side. Connect each consecutive midpoint with a segment. What are the properties of the shape formed by joining the midpoints? Does the resulting shape depend on the type of quadrilateral (e.g. convex & concave)? How does the area of the new figure compare to the area of the original quadrilateral?

Will the new figure ever be non-convex?

No matter if the quadrilateral is concave, convex or imbedded, the shaped formed is a parallelogram.  The reason is because the diagonal stay parallel to the midpoint connectors.

The areas formed are in a ratio of1:2.  See the GSP sketch above.

In make the three conclusions below, I drew a square and created the quadrilateral using circles.    I then manipulated this quadrilateral until I drew some conclusions about the square.

In order to form a square, the diagonals must be perpendicular and congruent.   I found this by looking at a kite.  Knowing the definition of a kite tells me that it is not a parallelogram but it does have a pair of congruent sides.  This way I could form triangles.  Low and behold, the triangles were congruent.

In order to form a rectangle, I would start with a quadrilateral whose diagonals are perpendicular to each other.

In order to form a rhombus the quadrilaterals diagonals must be congruent.

In order to form a square, the diagonals must be perpendicular and congruent. Which makes sense since a square is both a rhombus and rectangle.