Parallelogram

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A parallelogram is a four-sided plane figure that has two sets of opposite parallel sides. Every parallelogram is a polygon, and more specifically a quadrilateral. Special cases of a parallelogram are the rhombus, in which all four sides are of equal length, the rectangle, in which the two sets of opposing, parallel sides are perpendicular to each other, and the square, in which all four sides are of equal length and the two sets of opposing, parallel sides are perpendicular to each other. In any parallelogram, the diagonals bisect each other, i.e, they cut each other in half.

The parallelogram law distinguishes Hilbert spaces from other Banach spaces.

It is possible to create a tessellation with any parallelogram.

The three-dimensional counterpart of a parallelogram is a parallelepiped.

The area of a parallelogram can be seen as twice the area of a triangle created by one of its diagonals. The area can also be computed as the magnitude of the vector cross product of two of its non-parallel sides.

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Proof that diagonals bisect each other

Image:Parallelogram.svg Prove that the diagonals of a parallelogram bisect each other.

(Prove that <math>E/\overrightarrow{A C}=1:1</math> and <math>E/\overrightarrow{B D}=1:1</math>)

Proof:

<math>\overrightarrow{A E}=k\overrightarrow{A C}</math>, k is an element of the real numbers

<math>\overrightarrow{A E}=k(\overrightarrow{A D}+\overrightarrow{D C})

\Rightarrow

\overrightarrow{A E}=k(\overrightarrow{A D}+\overrightarrow{A B})</math> since <math>\overrightarrow{D C}=\overrightarrow{A B}</math>

<math>\overrightarrow{A E}=k\overrightarrow{A D}+k\overrightarrow{A B}</math>

since E,D,B are collinear, by the division-point theorem,

k + k = 1

2k = 1

k = 0.5

sub k = 0.5 into:

<math>\overrightarrow{A E}=k\overrightarrow{A C}</math>

<math>\overrightarrow{A E}=(0.5)\overrightarrow{A C}</math>

<math>\overrightarrow{A E} / \overrightarrow{A C}=0.5</math> (the ratio of AE to AC is 1:2)

also sub k = 0.5 into:

<math>\overrightarrow{A E}=k\overrightarrow{A D}+k\overrightarrow{A B}</math>

<math>\overrightarrow{A E}=0.5\overrightarrow{A D}+0.5\overrightarrow{A B}</math>

by the division-point theorem,

<math>E/\overrightarrow{D B}=1:1</math>

Image:Parallelogram2.PNG by adding the division ratios to the parallelogram, we see that E divides both diagonals in the ratio 1:1, and E bisects AC and BD.

Therefore, the diagonals of a parallelogram bisect each other.

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