A construction is an accurate drawing made with only two tools: a straight edge (an unmarked ruler, for drawing straight lines) and a pair of compasses (for drawing circles and arcs of a fixed radius). No measuring of angles, no guessing — the geometry does the work, so the result is exact.

The perpendicular bisector of a segment

Given a segment AB, the perpendicular bisector is the line that cuts it exactly in half and meets it at a right angle. To construct it:

• Open the compasses wider than half of AB.
• With that radius, draw arcs from A (above and below the line).
• Keeping the same radius, draw arcs from B; they cross the first pair at two points.
• Draw the straight line through those two crossing points.

That line is the perpendicular bisector. It works because the two crossing points are each the same distance from A as from B — and in fact every point on the line is equidistant from A and B.

Step through the construction:

The angle bisector

The same idea splits an angle in half. Given an angle at a vertex V:

• Put the compass point on V and draw an arc that crosses both arms of the angle.
• From each of those two crossing points, draw equal arcs in the middle of the angle so they meet.
• Draw the line from V through that meeting point: it splits the angle into two equal halves.

A triangle from three given sides

Compasses also build a triangle when you are told its three side lengths (this is the SSS case). Draw the base with the straight edge. Then set the compasses to the second side length and arc from one end of the base; set them to the third side length and arc from the other end. Where the two arcs cross is the apex — join it to both ends.