# 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.

# Diédrico System: Distance from a point to a line

We can define the distance from a point P to a line r as the smallest of the distances from the point P to the infinite points on the line r. To determine this distance must obtain the line perpendicular to the line r from the point P and get their point of intersection I. The distance d from P to R is the minimum distance from this point to the line r.

This problem can have two different approaches to determining the solution sought.

# System dihedral: Fundamentals of auxiliary projections, changes in plane

To represent an object in the dihedral system usually use the projections on the three planes of the reference trihedron, as we have seen when studying the fundamentals of dihedral system.

In general it is sufficient to use only two of the three possible planes, It is represented for example by a straight projections on the horizontal plane and the vertical. Sometimes it may be desirable, or even necessary, obtain new projections in different directions projection, in which case the call her “auxiliary projections” .

## 25 April, 2017

We have defined the ellipse as the “locus of centers circumferences, through a focus, They are tangent to the focal circumference of the other focus center”.

This definition allows us to approach the study of the conic by applying the concepts studied to solve the problems of tangents and, en particular, reducing them to the fundamental problem of tangents.

This circumference will link with another whose radius is half the radius of the focal, and its center is the taper. We call this circumference “Head circumference”.

# 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.

# Diédrico System: Distance from a point to a plane

We can define the distance from a point P to a α as the smallest of the distances from the point P up to the infinite points of the plane α. To determine this distance we get the straight perpendicular to the plane α from the point P and I get your point of intersection. The distance from P to I will be the minimum distance to the plane α.

# Perpendicular to a plane

One of the basic problems that we learn by studying the systems of representation are those in which there are elements that are perpendicular to other. All the problems of determining distances make use of these concepts.

Let's see how to determine the line perpendicular to a plane in dihedral system working directly on the main system projections.

# 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.