Vector geom.: If the midpoints of the consecutive sides ...

[Guest]

If the midpoints of the consecutive sides of any parallelogram are connected by striaght lines, prove by using vectors that the resulting quadrilateral is a parallelogram.

thank you in advance

 

[soroban]

Re: Vector geometry question

Hello, americo74!

If the midpoints of the consecutive sides of any parallelogram are connected by striaght lines,
prove by using vectors that the resulting quadrilateral is a parallelogram.

                   A       P           B
                   *- - - -*- - - - - -*
                  /     *    *        /
                 /   *         *     /
                / *              *  /
             S *                   * Q
              /  *              * /
             /     *         *   /
            /        *    *     /
         D-*- - - - - -*- - - -* C
                       R

We have parallelogram \(\displaystyle ABCD\) with midpoints \(\displaystyle P,\,Q,\,R,\,S.\)
Draw diagonal \(\displaystyle DB.\)

We have: \(\displaystyle \,\vec{SP} \:=\:\vec{SA}\,+\,\vec{AP} \:=\:\frac{1}{2}\vec{DA}\,+\,\frac{1}{2}\vec{AB}\:=\:\frac{1}{2}\left(\vec{DA}\,+\,\vec{AB}\right) \:=\:\frac{1}{2}\vec{DB}\)
. . Hence: \(\displaystyle SP\,\parallel\,DB\,\) and \(\displaystyle \,|SP|\,=\,\frac{1}{2}|DB|\)

We have: \(\displaystyle \,\vec{RQ}\:=\:\vec{RC}\,+\,\vec{CQ}\:=\:\frac{1}{2}\vec{DC}\,+\,\frac{1}{2}\vec{CB}\:=\:\frac{1}{2}\left(\vec{DC}\,+\,\vec{CB}\right)\:=\:\frac{1}{2}\vec{DB}\)
. . Hence: \(\displaystyle \,RQ\,\parallel\,DB\,\) and \(\displaystyle \,|RQ| \,= \,\frac{1}{2}|DB|\)

Then we have: \(\displaystyle \,SP\,\parallel\,RQ\,\) and \(\displaystyle \,|SP|\,=\,|RQ|\)

Theorem: if two sides of a quadrilateral are parallel and equal,
. . . . . . . .the quadrilateral is a parallelogram.

Therefore, \(\displaystyle PQRS\) is a parallelogram.