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Metric geometry : Homotecia

Transformaciones - homotecia

Transformaciones - homotecia

The homotecia is a transformation homográfica that maintains the relationships between each pair of measurement segments or homologous homothetic.

Preserves the parallelism between a line and its transformed, so determines and maintains similar figures angular relations (is in accordance).

Its main application is the determination of geometry problems with area ratios in similar figures; It is also useful for solving some exercises tangencies.

Two similar figures having the same shape and different area



It is based on the concepts of similarity that we saw in the Thales theorem; is not involutive transformation and can not have double elements except the center. It belongs to the group of affine transformations.

Transform definition

Dilation is a processing center. This means that a transformed point and are aligned with the center of dilation or likeness, analogously to the transformation known as investment will subsequently.

The relationship between the relative positions of each point and its transformed respect homothetic center is based on the concept of similarity.

Given a center “H“, and a pair of homologous points “P” and “P’“, the ratio of distances these points homothetic center is constant and is called reason homotecia.

HP / HP’ = HQ / HQ’ = HT / HT’ = K

Circunferencias homotéticas

Homothetic circles

Centers homotecia between the circles

Relate this transformation by two circles is of particular interest for its application in problems of tangents, and for the subsequent study of another transformation: investment.

Assuming that two circles are homothetic, the points on parallel radios must be homologous. Depending on the direction of the radius of positive reason we transformations (the two radios in the same direction) the negative (different sense). The positive centers, H , and negative, H-, must lie on the lines joining each pair of homologous points (A-A’) and the line joining the centers of the circumferences since they are also homothetic.

Centros de homotecia de dos circunferencias 1

Homotecia centers of the circles 1

We can see how in any particular positions of the centers of dilation can be located on their own circles, as is the case in which they are tangent to each other.

Centros de homotecia de dos circunferencias 2

Homotecia centers of the circles 2

If one is inside the other will also see the other homothetic center is interior to both circunferencis.

Centros de homotecia de dos circunferencias 3

Homotecia centers of the circles 3

Applying dilation problems of tangency

One of the possible applications of this transformation is the determination of circles with tangency conditions on two straight.

Suppose the following year:

Determine the circles tangent to two lines and passing through a point P

Homotecia - Problema de tangencias

Homotecia - Problem tangencies

If we assume that the point of intersection of the tangent lines is a center of dilation, H, we can convert the circumference we seek with any reason in another circle to be tangent to said straight. To perform this transformation within either choose for this new circumference

Homotecia - Problema de tangencias planteado

Homotecia - Problem raised tangencies

Point P point must have a counterpart, P‘, in the new circumference. This point will be at the intersection of this circle and the straight side r passing P and the center H de homotecia (Note that there may be another point of intersection of r with c’, valid for a second solution).

Homotecia - Problema de tangencias solucionado

Homotecia - Problem solved tangencies

The center of the circle determined by obtaining the solution the radius passing homolog P’, that passes through the point P and will be parallel to the anterior.

Metric geometry

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