# 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.

# 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.

# 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.

# How to create a 3D PDF for documentation and education

Current technology allows us to generate documents with rich content. In this case we will see how you can incorporate a 3D model to a document format “PDF”, retaining the three-dimensional model information, allowing us to change your display interactively.

# 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).

# Fall line

By studying the true magnitude of a line we saw that we could turn calculate the angle of this line with respect to a projection plane, namely, its slope.

In a plane we can determine endless lines with different direction contained therein. One of these lines form the maximum angular condition with respect to the projection plane.

# 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: Dynamic construction of a Tetrad of points [Geogebra]

Application “Geogebra” It allows you to develop dynamic constructions in which we can modify the position of the elements forming it, keeping the geometric constraints of these figures, allowing the invariants of the same show. This tool can be a valuable aid for students.

Professor Juan Alonso Alriols collaborated in the introduction of this tool in the teachings of “Graphic Expression” at the Polytechnic University of Madrid, providing examples of high interest. You can see an example of his work in the “Dynamic construction of double reason for four points” accompanying this entry, that has added a driver text for use in our classes.

# Geometría proyectiva: Construction of quadruples of points

We have seen the definition of ordered quadruples of elements, characterizing rectilinear some four points or four straight from a bundle of planes through a value or characteristic, result for the ratio of two triads determined by such elements.

We then consider the problem of obtaining, given three elements belonging to a same form of first category, series or beam, get a fourth element that determines a Tetrad of particular value.

# Metric geometry: Loci. Solución I (Selectivity 2014 – B1)

Vamos a resolver el problema de determinar un cuadrado, cuyos vértices se encuentran sobre elementos geométricos dados.
En particular fijaremos los correspondientes a una de sus diagonales sobre una recta, otro de los vértices en una recta diferente y el cuarto vértice sobre una circunferencia.