Some loci are elementary and serve to define geometric shapes known, while others require elaborate processes determining.
So, for example, the locus of points in the plane whose distance from a fixed point is constant is a circle with center point and radius referred to the distance given.
Relations in the triangle
The direct application of Pythagorean theorem we can obtain some loci high interest in the development of advanced theorems metric geometry.
The figure has the triangle ABC and have obtained, on the side “to“, the midpoint “M” and walk-up “H” to determine their height “h” from the apex “A“. This allows the determination of three triangles (a right angle) we can relate to each other for two major loci.
The triangles to which we refer are:
As shown in Figure, three triangles share the side “AH” as one of its legs, and the other leg is on the side “to”, base, Triangle; Triangles are longer than the side “AH” is the height of the triangle and thus is perpendicular to said base.
Applying theorem Pythagoras, we obtain the following three ratios:
adding the first two have the sum of two squares
whereas if we subtract one another have the difference of two squares
Locus of points whose difference of squares of distances from two fixed points is constant.
Let's see how we can use the above relations to determine the locus of points in the plane that meet the difference of the square of its distance from two fixed points is constant. This theorem we determine may be stated as follows:
The locus of points whose difference of the squares of the distances from two fixed points B and C is a constant quantity k is a line orthogonal to BC whose distance from the midpoint of BC is d = K/2BC.
Suppose that one of the points of the plane that fulfills this condition is the vertex “A” Triangle ABC, and fixed points to which we refer are “B” and “C“.
Locus of points whose sum of squares of distances from two fixed points is constant.
Expression obtained for the sum of squares:
it follows that, being “to” constant, so that it is the expression, must be the value “m” the median also a fixed value, thus concluded that the locus must be a circle of radius the value of the median.