 #### What are the principles of accurate construction?

When we speak of constructions in mathematics, we usually referred to those completed by compass and straightedge, independent of any specific unitary measure.

Let us consider the construction of a perpendicular bisector. Starting with a segment AB we construct two circles with equal radius centred at A and B. As long as the radius of the circles exceeds ½AB, the circles will intersect and by symmetry it is clear that the line drawn between the two points of intersection will bisect AB and will do so at right angles. In practice it is not necessary to draw complete circles, only a pair of arcs above and below the line are necessary to locate the points of intersection.

To bisect an angle a similar approach is used. Given an angle BAC, mark off two arbitrary equal lengths AX and AY using a compass. Then construct two circles radius XA and YA. Clearly the process of drawing circles XA and YA, allow the construction of the perpendicular bisector of XY, which is, by symmetry, the angle bisector of YAX and BAC.