Perturbed

[stoutfellow]

OK, here's the way I think about quadrilaterals: every quadrilateral arises by perturbing a parallelogram. Take a parallelogram PQRS. Move the opposite vertices P and R a certain distance, in a certain direction, and move the other two vertices the same distance in the opposite direction. The result is a new quadrilateral, a perturbation of the original parallelogram. Given a general quadrilateral, there's exactly one way to represent it as a perturbation; you can recover the parallelogram in this way. Let Z be the center of gravity of the quadrilateral. (If you use coordinates, Z = (A+B+C+D)/4. It's also halfway between the midpoints of the diagonals AC and BD.) If you reflect the midpoint M_AC of the diagonal AC in the midpoint M_AZ of the segment AZ, you'll get the first vertex of the parallelogram. Similarly, reflect M_BD in M_BZ, M_CA in M_CZ, and M_DB in M_DZ to get the other three vertices of the parallelogram.

Most of the classes of quadrilateral I've worked with involve either the base parallelogram by itself, or the relationship between the perturbation and various special lines of the parallelogram. More about that later; but this construction is key to the way I study quadrilaterals.

(I tried to insert a pair of pictures illustrating the constructions, but it didn't seem to work. Sorry.)