The humble parallelogram depends on Euclid's most controversial assumption
Everyone learns that a parallelogram's opposite sides match in length. What schoolbooks rarely mention is that this cannot be proved without Euclid's contested parallel postulate. The shape also carries a surprising list of hidden talents, from bisecting lines to squares that appear around it.
The name is Greek for a shape of parallel lines: a simple four-sided figure with two pairs of parallel sides. From that alone, and only with the parallel postulate or an equivalent, it follows that facing sides are equal and facing angles are equal. A four-sided figure with just one parallel pair is a trapezoid in American usage and a trapezium in British usage. The rectangle, rhombus and square are all special parallelograms. The old term rhomboid, for one with unequal sides and no right angles, has faded from mathematics but lingers in biology, as in the rhomboid muscles.
There are many ways to recognise one. A simple quadrilateral qualifies if both pairs of opposite sides are equal, or both pairs of opposite angles, or if one pair of sides is both parallel and equal. It also qualifies if each diagonal splits it into two congruent triangles, or if the squares of its four sides add up to the squares of its two diagonals, a result called the parallelogram law. Another test: the total distance from any interior point to the four sides stays constant, an extension of Viviani's theorem.
Its area is base times height, which you can see by slicing a right triangle off one end and sliding it to the other to form a rectangle. It also equals twice the area of the triangle cut off by a diagonal, or the size of the cross product of two neighbouring sides. The two diagonals split it into four triangles of equal area, and any line through the centre divides it in half.
Some facts are more surprising. Among convex polygons, only a parallelogram cannot fit inside any triangle smaller than double its own area. Build a square on each side and the four centres form another square. A parallelogram has rotational symmetry of order two, and any non-degenerate affine transformation turns one parallelogram into another.
Source: Parallelogram
