Math Forum - Project of the Month, February 1997
#### A Math Forum Project

# February POM - Winner

## Justin Lam

Grade 8

Sequoia Middle School, Pleasant Hill, California

**From: quan.lam@ucop**

Let the quadrilateral be ABCD.
Let E, F, G, H be the midpoints of sides AB, BC, CD, DA respectively.
AC and BD are the two diagonals.

Since E and F are midpoints of Triangle ABC, EF//AC.
Similarly, HG//AC because H and G are midpoints of Triangle DAC.
So, EF//HG.

Using the same argument for Triangles ABD and CBD, we have EH//BD and FG//BD.
So, EH//FG.

Therefore, for any quadrilateral, the quadrilateral formed by connecting
the consecutive midpoints must be at least a PARALLELOGRAM with the
length half the length of the corresponding diagonals.

- If the two diagonals AC and BD are of equal length, then the
parallelogram would be equilateral or a RHOMBUS.
- If the two diagonals are perpendicular to each other, then the
parallelogram would be a RECTANGLE.
- Of course, if the diagonals are both equal length and perpendicular
to each other, the parallelogram would be a SQUARE.

To answer your questions:

- Rectangle - the parallelogram is equilateral or a Rhombus (case 1)
- Kite - it is a rectangle (case 2)
- Isosceles Trapezoid - Rhombus (case 1)
- Square - Square (case 3)
- Rhombus - Rectangle (case 2)

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