In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are congruent. The three-dimensional counterpart of a parallelogram is a parallelepiped.
In a vector space, addition of vectors is usually defined using the parallelogram law. The parallelogram law distinguishes Hilbert spaces from other Banach spaces.
To prove that the diagonals of a parallelogram bisect each other, first note a few pairs of equivalent angles:
Since they are angles that a transversal makes with parallel lines and .
Also, since they are a pair of vertical angles.
Therefore, since they have the same angles.
From this similarity, we have the ratios
Since , we have
bisects the diagonals and .
The area formula,
can be derived as follows:
The area of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles. The area of the rectangle is
and the area of a single orange triangle is
Therefore, the area of the parallelogram is
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