Editing Parallelogram (section)

==Characterizations==
A [[simple polygon|simple]] (non-self-intersecting) [[quadrilateral]] is a parallelogram [[if and only if]] any one of the following statements is true:<ref>Owen Byer, Felix Lazebnik and [[Deirdre Smeltzer]], ''Methods for Euclidean Geometry'', Mathematical Association of America, 2010, pp. 51-52.</ref><ref>Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, p. 22.</ref>
*Two pairs of opposite sides are parallel (by definition).
*Two pairs of opposite sides are equal in length.
*Two pairs of opposite angles are equal in measure.
*The [[diagonal]]s bisect each other.
*One pair of opposite sides is [[Parallel (geometry)|parallel]] and equal in length.
*[[Adjacent angles]] are [[supplementary angles|supplementary]].
*Each diagonal divides the quadrilateral into two [[congruence (geometry)|congruent]] [[triangle]]s.
*The sum of the [[Square number|square]]s of the sides equals the sum of the squares of the diagonals. (This is the [[parallelogram law]].)
*It has [[rotational symmetry]] of order 2.
*The sum of the distances from any interior point to the sides is independent of the location of the point.<ref>Chen, Zhibo, and Liang, Tian. "The converse of Viviani's theorem", ''[[The College Mathematics Journal]]'' 37(5), 2006, pp. 390–391.</ref> (This is an extension of [[Viviani's theorem]].)
*There is a point ''X'' in the plane of the quadrilateral with the property that every straight line through ''X'' divides the quadrilateral into two regions of equal area.<ref>Problem 5, ''2006 British Mathematical Olympiad'', [https://artofproblemsolving.com/community/c6h63970p381087].</ref>
Thus, all parallelograms have all the properties listed above, and [[Converse (logic)|conversely]], if just any one of these statements is true in a simple quadrilateral, then it is considered a parallelogram.