# Investment: Table mental gymnastics for determination of elements with angular conditions

We have already used one “Table Mental Gymnastics” to study investment: a set of exercises that serve to stimulate thinking, develop and maintain an agile mind, automate processes calculation and analysis etc..

We now propose to raise a similar set of problems but aimed at obtaining solutions to basic problems of geometry. In this case we will raise finding circumferences passing through a given point and angular meet conditions on two circumferences.

# Learning Path Metric Geometry

In addressing the study of a science we can follow different paths that lead to learning. Chaining concepts linked to each other allow us to generate a mental representation of abstract patterns, facilitating their assimilation and subsequent application in problem solving.
In these pages two images that summarize a possible strategy or sequence of progressive incorporation of the basics of this branch of science in the education of our students are proposed.

# Investment: Table mental gymnastics processing elements

What is a table of mental gymnastics? We can say that is a set of exercises that serve to stimulate reasoning, develop and maintain an agile mind, automate processes calculation and analysis etc..
In the subjects of geometry we can propose a problem and make slight variations to any of the data. Variability problem will create families of exercises that emphasize one or more concepts of interest.

# Reversing a point. 10 constructions for obtaining [I- Metrics]

One recommendation I always do my students is to try to solve the same problem in different ways, instead of many times the same problems with almost similar statements.

We see a problem with metric or projective approaches in each case.

In one of my last classes we propose are obtaining the inverse of a point, an investment in the center and power is known. The proposed statement was as follows:

Since the square in Figure, in which one vertex is the center of inversion and the opposite vertex is a double point, determining the inverse of the point A (adjacent vertex).

# Geometría proyectiva: Obtaining conical shafts from two pairs Diameters Polar Conjugates

A conical axes are those conjugates polar diameters are orthogonal to each.

We recall that two polar conjugate diameters, necessarily pass through the center O of the conical, are the polar two points unfit (located at infinity) that they are conjugated, namely, the polar of each of these points contains the other.

These pairs of elements determine an involution of diameters (polar) Conjugates will be defined when two pairs of beams know and their homologues.

# Conical defined by the two foci and a tangent

We have solved the determination of a conic defined by the two foci and focal point by the circumference of the conical.

A problem using identical concepts is determining a known conic its foci and their tangents. We will see this problem in the case of an ellipse.

# Conical defined by the two foci and a point

One of the first problems we can solve based on the definition of conic as “locus of the centers of circumferences passing through a fixed point (focus) which are tangent to a circumference (focal circle centered the other focus)” It is the determination of the tapered from the two foci and a point.

The classic definition will be determined as the vertices A1 and A2 of the conical obtained.

# 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.

# 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.