Question 1206128
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Let the coordinates of the vertices of the quadrilateral be<br>
A(2a,2b)
B(2c,2d)
C(2e,2f)
D(2g,2h)<br>
The midpoints are<br>
AB: P(a+c,b+d)
BC: Q(c+e,d+f)
CD: R(e+g,f+h)
DA: S(g+a,h+b)<br>
Opposite sides PQ and RS have the same slope:
PQ: {{{((b+d)-(d+f))/((a+c)-(c+e))=(b-f)/(a-e)}}}
RS: {{{((f+h)-(h+b))/((e+g)-(g+a))=(f-b)/(e-a)=(b-f)/(a-e)}}}<br>
And opposite side QR and PS have the same slope:
QR: {{{((d+f)-(f+h))/((c+e)-(e+g))=(d-h)/(c-g)}}}
PS: {{{((b+d)-(h+b))/((a+c)-(g+a))=(d-h)/(c-g)}}}<br>
Both pairs of opposite sides are parallel, making the quadrilateral a parallelogram.<br>