# Loci: Difference of squares of distances from two fixed points The study of different loci that appear in the most common graphical models to understand and structure the graphic constructs used to solve many classical problems.

Given two fixed points, B and C in Figure, trying to determine the positions that can occupy the point A so that the difference between the square of the distance from A to these points is constant. To determine this locus we use the Pythagorean theorem. Triangles seek and will relate the length of its sides (distance between the vertices) through this celebrated theorem. In Figure assume that B and C are fixed points, and A belongs to the locus sought. Distance “to” between B and C is a constant value, being unchanged B and C two fixed points. If it is determined the midpoint M This side and the point H from the perpendicular A by BC, get up h and median m Triangle ABC.

Applying Pythagoras to triangles ABH and AHC we: We relate the squares of the sides of triangles (sought distances). Subtracting one equation to the other will: This equation tells us that if we want the difference of squares is constant, the product 2ad should be and, as to is a constant value, segment d should remain unchanged.

Geometrically point must remain fixed H and therefore the point A, which lies on the height of the triangle, should permenecer on a line perpendicular to BC passing through H.

The locus of points whose difference of squares of distances from two fixed points is constant, is a line perpendicular to the segment that determine the fixed points.

This locus is of great interest for the study of radical axis of two circles.

Metric Geometry