#### Winning Solution

Jennifer Burrows, Amanda Taplett

When you connect the consecutive midpoints of a quadrilateral, another quadrilateral is formed inside of it. We will call the original figure the mother figure and this new figure the daughter figure. To determine what the daughter is, you must examine the diagonals of the mother. It is possible to use the diagonals of the mother because the diagonals are parallel to the sides of the daughter.

The type of quadrilateral that is formed can either be a rhombus, a rectangle, or a square, but it will always be a parallelogram. This is because when the midpoints are connected to form the sides of the daughter figure, each side of the mother figure is bisected. Each newly formed side will be parallel to a diagonal of the mother. Two of the newly formed sides are parallel to the same diagonal and therefore are parallel to each other. Along with the other two sides of the daughter that are parallel to the other diagonal of the mother, a parallelogram is formed.

Whether the daughter quadrilateral is a regular parallelogram or a rhombus, rectangle, or square is dependent on the diagonals of the mother quadrilateral. If the diagonals of the mother are congruent, then the daughter will be a rhombus. If the diagonals are perpendicular, the product will be a rectangle. Finally, if the diagonals are both equal and perpendicular, the daughter will be a square. If the mother's diagonals are neither perpendicular nor equal, the daughter is an ordinary parallelogram.

The midpoints, when joined, create four triangles. These triangles determine the diagonals of the figure. The diagonals are always parallel to the sides of the daughter figure. This is because the diagonals make similar triangles to those of the ones created by the midpoints. For example, if the triangles created by the midpoints are isosceles, the triangles formed by the diagonals will be as well. This is why, if we know about the diagonals of the mother figure, we are able to discover the kind of quadrilateral the daughter will be. If the diagonal forms the altitude of the triangle, then the daughter is a square. If it forms a median, then it is a rhombus. If the diagonal forms both the median and the altitude in the same place, then the daughter will be a rectangle.

Therefore, an isosceles trapezoid or a rectangle would have a rhombus for a daughter. A rhombus or kite's daughter would be a rectangle. A square's daughter would always be a square.

By the diagonals, the daughters of a quadrilateral are able to be calculated. In conclusion, we discovered that the daughter of the mother figure formed similar triangles which enabled us to determine the characteristics of the daughter. With this information, we are able to decide what kind of quadrilateral the daughter will be and save ourselves a great deal of work.

COMMENTS: Jennifer and Amanda stated early on that the resulting quadrilateral was always a parallelogram, and explained why. They could have talked about the theorem that says the same thing, but it's certainly not wrong the way it is. Where they went beyond the second submission, however, is that they explored the special cases, and studied what it would be if it were a square, parallelogram, rhombus, etc., and talked about why. Very thorough!

#### Honorable Mention

Hello, our names are Susie Taylor and Colleen Cunningham. We are sophomores, in Mr. Detzel's Honors Geometry class, at Shaler High School. Here is our solution to the Problem of the Month.

In a quadrilateral, (convex or nonconvex), when you connect the midpoints (consecutively), you get a shape in the middle.

Using the Midline Theorem, the segment between the midpoints of two sides of a triangle is parallel to the third side and half as long. To make this theorem work, you must draw in one diagonal of the quadrilateral. This would form two triangles. Thus, since both segments are parallel to the diagonal, the segments are parallel to each other by Transitivity. The shape that you would end up with will always be a parallelogram.

We found this to be true with specific types of quadrilaterals: squares, Rectangles, Trapezoids, Parallelograms, Rhombuses, Kites, Diamonds, Arrows, and other different quadrilaterals.

Thanks for your time, and we would really appreciate any suggestions you may have.

```                                   Sincerely,
Susie and Colleen
```
COMMENTS: I especially like the fact that they stated the theorem they used, and explained it right off the bat. However, they might have done a little more exploring with the other shapes - while the parallelogram still holds, what's interesting about the other shapes?