prove diagonals of parallelogram bisect each other using vector methods (1 Viewer)

[RohitShubesh21]

hello sirs, I found that diagonals equal to each other but not sure if that mean it bisect, can someone show proper proof thank u sirs

 

[notme123]

i think you want to show each side of the bisected diagonal is the same vector

 

[Drongoski]

The diagonals of a parallelogram bisect each other. So in a //gram ABCD, the diagonals AC and BD bisect each other. What this means is that AC cuts BD into 2 equal halves and BD cuts AC into 2 equal halves.
In an arbitrary quadrilateral ABCD, diagonal AC may cut diagonal BD into 2 equal halves, but BD will not cut AC into 2 equal halves unless ABCD is a //gram; in which case it is no longer an arbitrary quadrilateral.


Let ABCD be a parallelogram. Therefore has 2 sides parallel and equal in length. Without loss of generality, let these be AD & BC; .: AD//BC.
Let the mid-point of AC and of BD be E and F respectively. Let the position vectors OA, OB, ... , OE, and OF be resp.:



That means the points E and F are the same point.
Therefore the mid-point of AC is the mid-point of BD

That means the diagonals of //gram ABCD bisect each other.

QED

Strictly speaking, I should start from the property: AB//DC and AD//BC without assuming opp sides are equal.