Geometry construction using a compass and straightedge
This page shows how to construct (draw) a
inscribed in a circle with a compass and straightedge or ruler. This is the largest hexagon that will fit in the circle, with each
touching the circle. In a regular hexagon, the side length is equal to the distance from the center to a vertex, so we use this fact to set the compass to the proper side length, then step around the circle marking off the vertices.
As can be seen in Definition of a Hexagon,
each side of a regular hexagon is equal to the distance from the center to any vertex.
This construction simply sets the compass width to that radius, and then steps that length off around the circle
to create the six vertices of the hexagon.
The image below is the final drawing from the above animation, but with the vertices labelled.
A,B,C,D,E,F all lie on the circle center O
AB = BC = CD = DE = EF
They were all drawn with the same compass width.
From (2) we see that five sides are equal in length, but the last side FA was not drawn with the compasses.
It was the "left over" space as we stepped around the circle and stopped at F.
So we have to prove it is congruent with the other five sides.
OAB is an equilateral triangle
AB was drawn with compass width set to OA, and OA = OB (both radii of the circle).