Within a circle or in congruent circles, congruent central angles have congruent chords.
Using this theorem, to construct a regular octagon we can draw eight congruent central angles and then draw eight chords — which will be the sides of the octagon. First, let's draw a circle with an arbitrary radius.
Next, we will draw a horizontal diameter through the center O.
Now we need to draw second diameter, which is a perpendicular bisector of the first diameter. To do that we should put the compass point on point A and draw an arc. The opening of the compass must be greater than 12AB — it must go further than the midpoint O. Otherwise, the arcs we draw will not intersect.
With the same compass setting, we will put the compass point on point B and draw another arc.
Let's draw a segment through the points of intersection so that the segment intersects the circle. We will label its points of intersection with the circle as C and D.
Next, we need to draw bisectors of the four right angles formed. Let's start with ∠ COB and ∠ AOD. To do this, we will put the compass point on the vertex O and draw an arc that intersects the sides of ∠ COB. For future reference, we will call the points of intersection M and N.
Next, we will put the compass point on point M and draw an arc near the center of the angle. We need to place the arc in the center so that it will intersect the arc that we create from point N.
With the same compass setting, we will draw the arc using point N. Remember, we need the two arcs to intersect.
Finally, we will draw a sector through O and the point of intersection of the arcs so that it intersects the circle at two places. This is the bisector of ∠ COB and ∠ AOD.
Similarly, we can draw a bisector of ∠ AOC and ∠ BOD.
We have constructed eight congruent central angles. Let's draw their chords, which by the Theorem 12-5, are also congruent.
