Reconstruct a quadrilateral ABCD given length of its sides and the length of the midline between the first and third sides (namely all the segments drawn in the given figure) using compass and straight edge.
Solution: Reflect $A,B$ and $E$ across $F$ to $A',B'$ and $E'$. Then $AEA'E'$, $BEB'E'$ and $ABA'B'$ are parallelograms. Also $AEE'B'$ is parallelogram. Let $G$ halves $AB'$, then since $F$ halves $EE'$ we see that $AEFG$ is also parallelogram, so we know the length of $FG$.
Now the construction:
- Draw green triangle $AB'D$ (we know all it sides)
- Draw blue triangle $DGF$ (we know all it sides)
- Draw quadrilateral $ABCD$.
