Graphic PIZiadas

Graphic PIZiadas

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Loci: Difference of squares of distances from two fixed points

lugares_geométricosThe 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.

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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