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**.

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