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The centers of four squares all constructed either internally or externally on the sides of a parallelogram are the vertices of a square. [8] If two lines parallel to sides of a parallelogram are constructed concurrent to a diagonal, then the parallelograms formed on opposite sides of that diagonal are equal in area. [8]
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.
An arbitrary quadrilateral and its diagonals. Bases of similar triangles are parallel to the blue diagonal. Ditto for the red diagonal. The base pairs form a parallelogram with half the area of the quadrilateral, A q, as the sum of the areas of the four large triangles, A l is 2 A q (each of the two pairs reconstructs the quadrilateral) while that of the small triangles, A s is a quarter of A ...
The midpoints of the sides of any quadrilateral (convex, concave or crossed) are the vertices of a parallelogram called the Varignon parallelogram. It has the following properties: Each pair of opposite sides of the Varignon parallelogram are parallel to a diagonal in the original quadrilateral.
The extended parallelogram sides DE and FG intersect at H. The line segment AH now "becomes" the side of the third parallelogram BCML attached to the triangle side BC, i.e., one constructs line segments BL and CM over BC, such that BL and CM are a parallel and equal in length to AH.
For a given parallelogram consider an arbitrary inner parallelogram having as a diagonal as well. Furthermore there are two uniquely determined parallelograms G F H D {\displaystyle GFHD} and I B J F {\displaystyle IBJF} the sides of which are parallel to the sides of the outer parallelogram and which share the vertex F {\displaystyle F} with ...
The upper figure shows that for a scalene triangle, the area of the parallelogram on the longest side is the sum of the areas of the parallelograms on the other two sides, provided the parallelogram on the long side is constructed as indicated (the dimensions labeled with arrows are the same, and determine the sides of the bottom parallelogram).
Any of the sides of a parallelogram, or either (but typically the longer) of the parallel sides of a trapezoid can be considered its base. Sometimes the parallel opposite side is also called a base, or sometimes it is called a top, apex, or summit. The other two edges can be called the sides.