# Conic as Locus Centers Circumferences Tangents

We have seen that the study of conic can be made from different geometric approaches. En particular, to start analyzing conic we have defined as the ellipse locus, we said that:

Ellipse is the locus of points in a plane whose sum of distances from two fixed points, called Spotlights, It has a constant value.

This metric definition of this curve allows us to address important study relating to the tangents circumferences, known as “Problem of Apollonius” in any of its versions. When we approach the study of the parabola or hyperbola return to reframe the problem to generalize these concepts and reduce problems “fundamental problem of tangents in the case straight”, or “fundamental problem of tangents in the case circumference”, namely, determining a circumference of a “Make corradical” a tangency condition.

# Metric geometry : Investment beam circumferences

Transformation through investment in geometric shapes grouped elements can be of interest to use the investment as a tool for analysis in complex problems. In this case study transforming “beams circumferences corradicales” through various investments that transform. Later these transformations need to solve the problem “Apolonio” (circumference with three tangency constraints) o la “Generalization of the problem of Apollonius” (circumferences with three angular restrictions).

# The robustness of dynamic geometric constructions with Geogebra: Polar of a point of a circle

The study of the disciplines of classical geometry can be reinforced by using tools that allow constructions that can be changed dynamically: variational constructions.
The tool “Geogebra” It will serve to illustrate these concepts and demonstrate the importance of detailed knowledge of geometric relationships to ensure the robustness of the buildings we use in geometric reasoning, as, sometimes, some constructions may lose their validity.

# projective center two beams [Interactive] [Geogebra]

a conical (punctual) It is the locus of the points of intersection of two projective beams.
This model has been verified with a variational model of projective shaft made with Geogebra.

# Projective axis of two series [Interactive] [Geogebra]

Projective geometry constructions made with tools to analyze their invariants are very useful for the study of this discipline of Graphic Expression. We will see one of these constructions made with the software “GeoGebra”, in particular for determining the projective axis of two projective series.

# Triangle geometry [Problem]

We have seen in studying the concept of power or theorems Hick and height metric relations between segments.

In these relationships, along with the Pythagorean Theorem segments are related by quadratic forms that can also be interpreted as areas (product of two lengths)

# Conical : Ellipse as locus

The study of conic can be made from different geometric approaches. One of the most used is the analysis that determined from planar sections in a cone of revolution.

From this definition it is possible to infer metric properties of these curves, plus new definitions of the same.

# The problem of the spin Center

A rotation in the plane is determined by its Center (spinning) and the rotated angle. This is equivalent to defining three simple data, two for the Center (coordinates “x” and “and”) and one for the value of the angle in degrees in any of the three systems of units that we use, centesimal degree, sexagesimal and radians.

Normally we tend to solve many direct problems in which there are twists in geometry. Give us a figure and we request that, with a true Center, revolve it with a certain angle. Less common is the reverse problem.

# To be Professor of drawing in high school you need a Master

To become Professor of technical drawing in secondary, What to do?

Many of my students have asked me what to do to be Professor of drawing, course that I teach at the University. The answer is always the same do teacher what? It is not the same be University professor who became an Institute Professor.

# Geometría proyectiva : Center of involution

We have seen how to determine the axis of an involution and, based on the concept of polar of a point with respect to two lines, possible Involutions which can be set from four points, with their respective shafts of involution, obtaining the autopolar triangle associated which are harmonious relations of the cuadrivertice full.

In this article we will continue to enhance these elements, in particular in the autopolar triangle vertices that will determine what is known as “Center of involution”.

# Projective Geometry: Autopolares triangles in Involutions in second-order series

Connecting four points of a conical proyectivamente by Involutions we determine the axis of involution of these proyectividades.

Given the four points needed to define an involution, We can ask many different Involutions can establish between them.