Loci

Circle:

All points on a circle have the same distance from its center point.

Ellipse:

The sum of the distances to the two focal points is the same for all points on the ellipse.

Equidistant Line:

All points that are equidistant from two specified points.

Parallels:

All points equidistant to a specified straight line.

Bisector:

All points equidistant to two specified lines.

Circle of Apollonius:

Every angle $v$ that spans a line segment. Added together, these angles create a circle.