bisecting a segment
You have a segment AB and you want its exact middle, without measuring its length and halving the number. Folding paper would do it, but here you have only compass and straightedge. The construction not only finds the midpoint but also throws in, for free, the line that cuts AB at a right angle through that midpoint.
Open the compass to more than half of |AB| — any width past the middle works. Centre it at A and draw an arc above and below the segment; without changing the width, centre it at B and draw a second pair of arcs. The two arcs cross at two points, one above the line and one below; call them M and N. Draw the line MN with the straightedge. It passes through the midpoint of AB and meets AB at a right angle: it is the perpendicular bisector. The reason is that every point equidistant from A and B lies on this line, and M and N were each built equidistant from A and B.
Bisecting a segment is one of the most reused constructions in all of geometry. The perpendicular bisectors of a triangle's three sides all pass through one point, the circumcentre, which is the centre of the circumscribed circle. It is also the first half of dividing a segment into equal parts and a building block for erecting perpendiculars.
From A and B swing arcs of equal radius (bigger than half of |AB|) above and below; they meet at M and N. Line MN crosses AB at its midpoint and is perpendicular to it — that is the perpendicular bisector.
Two equal-radius arcs from each end pin down both the midpoint and the perpendicular at once.
The compass width must exceed half of |AB|, or the arcs never meet. Note that the line you get is the perpendicular bisector — the midpoint is just where it crosses AB.