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In Euclidean geometry, a parallelogram is a simple

  • In Euclidean geometry, a parallelogram is a straightforward (non-self-converging) quadrilateral with two sets of parallel sides. The inverse or confronting sides of a parallelogram are of equivalent length and the inverse edges of a parallelogram are of equivalent measure. The harmoniousness of inverse sides and inverse points is an immediate outcome of the Euclidean parallel hypothesize and neither one of the conditions can be demonstrated without speaking to the Euclidean parallel propose or one of its proportional plans. 

  • By correlation, a quadrilateral with only one sets of parallel sides is a trapezoid in American English or a trapezium in English. 

  • The three-dimensional partner of a parallelogram is a parallelepiped. 

  • The historical background (in Greek παραλληλ-όγραμμον, a shape "of parallel lines") mirrors the definition.A straightforward (non-self-crossing) quadrilateral is a parallelogram if and just if any of the accompanying explanations is true:[2][3] 

  • Two sets of inverse sides are equivalent long. 

  • Two sets of inverse edges are equivalent in measure. 

  • The diagonals cut up each other. 

  • One sets of inverse sides are parallel and equivalent long. 

  • Contiguous points are supplementary. 

  • Every askew partitions the quadrilateral into two consistent triangles. 

  • The aggregate of the squares of the sides measures up to the entirety of the squares of the diagonals. (This is the parallelogram law.) 

  • It has rotational symmetry of request 2. 

  • The aggregate of the separations from any inside indicate the sides is autonomous of the area of the point.[4] (This is an augmentation of Viviani's hypothesis.) 

  • In this manner all parallelograms have every one of the properties recorded above, and on the other hand, if only one of these announcements is valid in a straightforward quadrilateral, then it is a parallelogram. 

  • Properties[edit] 

  • Inverse sides of a parallelogram are parallel (by definition) thus will never converge. 

  • The range of a parallelogram is double the territory of a triangle made by one of its diagonals. 

  • The region of a parallelogram is additionally equivalent to the extent of the vector cross result of two contiguous sides. 

  • Any line through the midpoint of a parallelogram cuts up the area.[5] 

  • Any non-worsen relative change takes a parallelogram to another parallelogram. 

  • A parallelogram has rotational symmetry of request 2 (through 180°) (or request 4 if a square). In the event that it additionally has precisely two lines of reflectional symmetry then it must be a rhombus or an elongated (a non-square rectangle). On the off chance that it has four lines of reflectional symmetry, it is a square. 

  • The edge of a parallelogram is 2(a + b) where an and b are the lengths of nearby sides. 

  • Not at all like some other raised polygon, a parallelogram can't be engraved in any triangle with not as much as twice its area.[6] 

  • The focuses of four squares all developed either inside or remotely on the sides of a parallelogram are the vertices of a square.

  • On the off chance that two lines parallel to sides of a parallelogram are developed simultaneous to a corner to corner, then the parallelograms framed on inverse sides of that askew are equivalent in area. 

  • The diagonals of a parallelogram partition it into four triangles of equivalent area.An automedian triangle is one whose medians are in an indistinguishable extents from its sides (however in an alternate request). I#n the event that ABC is an automedian triangle in which vertex A stands inverse the side a, G is the centroid (where the three medians of ABC meet), and AL is one of the expanded medians of ABC with L lying on the circumcircle of ABC, then BGCL is a parallelogram. 

  • Varignon parallelogram 

  • The midpoints of the sides of a self-assertive quadrilateral are the vertices of a parallelogram, called its Varignon parallelogram. On the off chance that the quadrilateral is arched or inward (that is, not self-crossing), then the range of the Varignon parallelogram is a large portion of the region of the quadrilateral. 

  • Digression parallelogram of an ellipse

  • For a circle, two breadths are said to be conjugate if and just if the digression line to the oval at an endpoint of one width is parallel to the next distance across. Every match of conjugate distances across of a circle has a comparing digression parallelogram, once in a while called a jumping parallelogram, framed by the digression lines to the oval at the four endpoints of the conjugate breadths. All digression parallelograms for a given oval have similar region. 

  • It is conceivable to remake an oval from any combine of conjugate distances across, or from any digression parallelogram.

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