Locus

The set of all points satisfying a condition.

The idea

A locus is the set of all points satisfying a stated condition — every point that satisfies it, and no point that does not. To name a locus, identify the figure those points form. Three examples recur: the points at a fixed distance $r$ from a fixed point form a circle of radius $r$; the points equidistant from two fixed points form the perpendicular bisector of the segment joining them; and the points at a fixed distance $d$ from a line form two lines parallel to it, one on each side.

A locus must contain every point meeting the condition. When the condition can be met on either side of a line, as in the last example, the locus has two pieces.

When a problem imposes two conditions at once, the points satisfying both form the intersection of the two loci: draw each figure separately, and take the points where they meet.

Ways to work on it

• Walkthrough. The classic loci: circle, perpendicular bisector, parallel lines.
• Practice. Name the shape a condition carves out.
• Hardest. Count the points where two loci intersect.

Not sure where to start? Take the ten-question placement test.