Prove that the quadrilateral formed by connecting the midpoints of a quadrilateral is a parallelogram.

Question

Given a quadrilateral $ABCD$, prove that the quadrilateral formed by its midpoints, $EFGH$, is a parallelogram.

Answer 1

Since EH is a midline of triangle ABD, $EH \parallel BD$. Likewise, $FG \parallel BD$ looking at triangle BCD. Therefore $EH \parallel BD \parallel FG$. Similarly you can show $EF \parallel HG$.

Answer 2

Hints: Let vectors $a,b,c,d$ go from an (arbitrary but fixed) origin to points $A,B,C,D$. Then

1. Express the vectors from the origin to the midpoints $E,F,G,H$, using $a,b,c,d$.
2. Conclude that $\vec{EF}=\vec{HG}$.