The sum of the squares of the 4 sides of a parallelogram equals that of the 2 diagonals
Not to be confused with Parallelogram rule (physics).
The sides of parallelogram ABCD are shown in blue and the diagonals in red. The sum of the areas of the blue squares equal that of the red ones.
In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. We use these notations for the sides: AB, BC, CD, DA. But since in Euclidean geometry a parallelogram necessarily has opposite sides equal, that is, AB = CD and BC = DA, the law can be stated as
If the parallelogram is a rectangle, the two diagonals are of equal lengths AC = BD, so
and the statement reduces to the Pythagorean theorem. For the general quadrilateral (with four sides not necessarily equal) Euler's quadrilateral theorem states
where is the length of the line segment joining the midpoints of the diagonals. It can be seen from the diagram that for a parallelogram, and so the general formula simplifies to the parallelogram law.
In mathematics, the simplest form of the parallelogramlaw (also called the parallelogram identity) belongs to elementary geometry. It states that the...
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