Is it possible to construct a hexagon using only a straightedge and a compas?

Answer 1

Yes

First construct an equilateral triangle:

Draw the base side of the triangle and label the ends A and B - this will be the first side of the hexagon.

Set the compass to the length of the side.

With the compass on one end of the line (point A), draw an arc to one side of the line (roughly near the middle).

With the compass on the other end of the line (point B), draw a second arc to intersect the first arc (call this point O) - this is the apex of the triangle.

Normally when constructing the triangle side OA and OB would be drawn in, however as a hexagon is being constructed only the location of O is needed. This point O will be the centre of the hexagon.

Now construct 4 further equilateral triangles:

For the first use OB as the base (the actual line is not needed, just its endpoints), construct its apex C and using the straight edge join B to C.

For the next, use OC as the base, construct its apex D and join C to D

Then use OD, construct apex E and join D to E

Finally using OE, construct apex F and join E to F.

The hexagon can now be completed by joining F to A.