Let ABCD be a quadrilateral.

Question

Let $ABCD$ be a quadrilateral. $E,F$ are midpoints of diagonals $AC,BD$. If $\vec{AB}-\vec{BC}+\vec{AD}-\vec{DC}=k\vec{FE}$, then $k$ is:

Accepted Answer

With position vectors $a,b,c,d$: $\vec{AB}-\vec{BC}+\vec{AD}-\vec{DC} = (b-a)-(c-b)+(d-a)-(c-d) = 2b+2d-2a-2c$.

$E=\dfrac{a+c}2$, $F=\dfrac{b+d}2$: $2b+2d-2a-2c = 4(F-E) = -4(E-F) = -4\vec{FE}$.

$k=-4$

Solutions · 1