Several standard locus problems yield circles. A point equidistant from two fixed points traces the perpendicular bisector (a line), but a point at a fixed distance from a point traces a circle. The locus of the midpoint of chords of a fixed length in a circle is a concentric circle. The locus of the midpoint of chords subtending a fixed angle at the centre is another concentric circle. The director circle (locus of perpendicular tangent pairs) is x^{2+y}^2=2$a^2$. The locus of a point with a constant ratio of tangent lengths to two circles is generally a circle (Apollonius circle). A circle touching a fixed line at a fixed point has its centre on the normal at that point.