Dissecting using a ruler and compass

Question

The problem is to cut a regular hexagon into parts that can be put together (without overlaps or wasting any parts) to make an equilateral triangle using only a ruler and compass (and scissors).

What is the smallest number of parts that will still let you achieve this?

http://mathworld.wolfram.com/Dissection.html shows a picture of a solution that uses 5 parts but I don't know if this can be done with a ruler and compass. I also have no idea how to prove that you can't do it in 4 parts.

Update. It seems that a solution by Harry Lindgren (1961) that uses only 5 pieces can indeed be done using ruler and compass. The question remains how to show it can't be done in 4 pieces. I am also interested to know if Lindgren's solution is unique in some sense.

Answer 1

In fact the solution shown by Wolfram Mathworld is a constructible five-piece solution, presumably Lindgren's.

Let each side of the hexagon be one unit, and label the hexagon $ABCDEF$ with $A$ at the vertex on the left side and the other vertices ordered clockwise. Then the cut from $A$ to point $G$ on side $BC$ measures half the side of the equilateral triangle, thus $\sqrt6/2$. Point $H$ on side $EF$ is at distance $\sqrt6/2×\sqrt3/2=3\sqrt2/4$ from the cut $AG$, and the cut from $H$ to $I$ on cut $AG$ makes angle $AIH$ which measures $60°$. The length of $HI$ measures $\sqrt6/2$, congruent with $AG$. The remaining cut from $I$ bisects angle $GIH$ and the remaining two cuts (from $D$ and $E$) meet at the center of the hexagon.