Midpoint
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
The straightedge buys nothing
Every point a compass and a straightedge can construct together can be constructed by the compass alone. The straightedge draws lines nobody needs; the compass does the work, and the proof that it does is an inversion performed with arcs.
One circle, and a straightedge
A straightedge alone cannot bisect a segment, so it cannot draw a parallel, so it can construct almost nothing. Draw one circle anywhere and mark its centre and everything a compass could ever have done becomes available — the circle is never needed again.
The compass that will not open
Fix the compass at one opening and never change it. That looks like a serious loss — a circle of a given radius through a given point is the compass's whole job — and it turns out to cost nothing at all, for reasons that are arithmetic rather than geometric.
Named alongside it
The objects these essays reach for when they reach for this one.
Constructible numberConstructionStraightedgeCircleCompassCross-ratioHarmonic conjugateIncidenceInversionOperation setParallelProjective geometry