,
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.
Argument
Reason
1
A,B,C,D,E,F all lie on the circle O
By construction.
2
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.
3
OAB is an equilateral triangle
AB was drawn with compass width set to OA,
and OA = OB (both radii of the circle).
5
m∠AOF = 60°
As in (4) m∠BOC, m∠COD, m∠DOE, m∠EOF are all &60deg;
Since all the central angles add to 360°,
m∠AOF = 360 - 5(60)
So now we have all the pieces to prove the construction
8
ABCDEF is a regular hexagon inscribed in the given circle
- From (1), all vertices lie on the circle
- From (20), (7), all sides are the same length
- The polygon has six sides.
-
Q.E.D
containing two problems to try.
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