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bisecting an angle
Splitting an angle exactly in two using only a pair of compasses and a straightedge is easy and the ancient Greeks knew very well how to do it. Put the point of the compass at a point O and draw a circle so that it cuts the two lines coming out from the angle. Call these intersection points A and B. Now put the point of the compass at A and draw an arc that follows within the opening of the angle. Without changing the radius at which the compass is set, move its point to B and draw another arc. Join the point where the two arcs cross, P, to O using the straight edge: angle POB is half of angle AOB.
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