# Metric Geometry

The study of the different geometries (Metrics, descriptive, projective) starts at the bachelor but finds its most abstract to the university levels.

The geometry (Latin geometry, which comes from the Greek language γεωμετρία, geo landmetria measure), is a branch of mathematics that deals with the study of the properties of geometric figures in the plane or space, like: points, straight, planes, polytopes (including parallel, perpendicular, curves, surfaces, polygons,polyhedra, etc.).(W)

These pages present a set of structured basics that allow a reasoned initiate us in beautiful study of this discipline, geometry, basis of understanding in mechanical engineering designs.

# Metric Geometry

The metric geometry is applied in Euclidean space is based on the theorems of Thales and Pythagoras.

The notions of the relationship between central and inscribed angles in a circle round the basics of learning to understand the Metric Geometry.

The lesson plans presented sequentially the set of concepts underlying geometric constructions. Add basic exercises that strengthen and settle these concepts.

All PPT files are in PowerPoint format

Preliminary Concepts (CP): Study of some theorems and elementary properties of classical geometry.

 Learning Path Metric Geometry

## Basic concepts in metric geometry

Notions of angles

Theorem of Thales

Pythagorean Theorem

Equation of the circle

Angles on the circumference : Central and inscribed

Arco able on a segment

Arco able on a segment : Example [I]

Arco able on a segment : Solución [I]

Arco able : Problema II

Arco able : Problema II Solución

## Potency to a circumference

Potencia de un punto respecto de una circunferencia

Generalization of the concept of "Power"

## Triangle geometry

Theorems height and leg

Triangle geometry: Problem 1

## loci of interest

Difference of squares of distances from two fixed points

## Geometric transformations

Geometric transformations : Correlations Vs homographies

Homotecia

Investment in the plane

Reversing a point (Metrics)

Inversion table

The problem of the spin Center

## Equivalence

Square equivalent

## Fundamental Problem Tangencies (PFT)

Problema fundamental de tangencias : PPr

The problem with football

Problema fundamental de tangencias : PPc

Determination of lines with angular conditions

Determining radio circumferences known angular conditions

Investment : Application to the resolution of problems and angular tangents

Apollonius and his ten problems

Problem of Apollonius : rcc

Problem of Apollonius : ccc

Sum / Difference of squares of distances from two fixed points

Make circles parabolic

Haz elíptico de circunferencias

Make hyperbolic circles

Generalization of the fundamental problem of tangents

Investment beam circumferences

Polar of a point P with respect to a circumference c Geogebra

## Problems

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

## Conical

Conical

Ellipse as locus

Conic as Locus circumferences Tangents

Conical defined by the two foci and a point

Conical defined by the two foci and a tangent

## Problems

### Loci

Selectivity 2014 – B1

Circunferencias con condiciones angulares

Solución

The problem with the pool table

The problem with the pool table: Solución

The problem of the two peoples and the bridge

The problem with football

Determinación de un segmento conocido su punto medio

Determinación de un segmento conocido su punto medio [Solución]

Enlace a Temas de Geometría Métrica elaborados for mis alumnos

## Basics

• LO-F-010. CP4-Power (CP_v14c4). Potencia de un punto respecto de una circunferencia. Training.

## Transformaciones

• LO-F-012. CP6-Equivalencia (CP_v14c6). Teoria y ejercicios.
• LO-F-013. CP7-Transformaciones (CP_v14c7). Teoria y ejercicios.

## Complementos

• LO-F-014. CP8-Complementos (CP_v14c8). Complementos de Geometria.