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

Corradicales circumferences Beams
Sum / Difference of squares of distances from two fixed points	Radical Axis
Getting Radical Axis	Corradicales circumferences Beams
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	conic metric: Head circumference
Conical defined by the two foci and a point	Conical defined by the two foci and a tangent

Problems
Loci
Selectivity 2014 – B1	Solución Selectividad 2014 – B1
Angularidad Básica
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-007. CP1-Thales (CP_v14c1). Thales theorem. Similarity of triangles. Preliminary Concepts. Training.
• LO-F-008. CP2-Pythagoras (CP_v14c2). Pythagoras. Triangle geometry. Training.
• LO-F-009. CP3-Angles (CP_v14c3). Angles on the circumference. Arco able. Training.
• LO-E-002. http://es.wikipedia.org/wiki/Teorema_de_Thales. Wikipedia: Thales.
• LO-E-003. http://es.wikipedia.org/wiki/Tri%C3%A1ngulos_semejantes. Wikipedia: Similarity. Similar Triangles.
• LO-E-004. http://es.wikipedia.org/wiki/Teorema_de_Pit%C3%A1goras. Wikipedia: Pythagoras. Pythagorean Theorem.

Corradicalidad

• LO-F-010. CP4-Power (CP_v14c4). Potencia de un punto respecto de una circunferencia. Training.
• LO-F-011. CP5-Eje Radical (CP_v14c5). Radical axis of two circles. Centro radical. 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.

Repaso acumulado

• LO-F-015. Construccion de Formas poligonales (apuntes_4001_formasPoligonales.PDF). Poligonos Apuntes y del Profesor de Triangulos Victorino Gonzalez García editados en la Sección de publicaciones de la EUITA. (UPM).
• LO-F-016. Preliminary Concepts (PDF). Apuntes del Profesor de Conocimientos preliminares Victorino Gonzalez García editados en la Sección de publicaciones de la EUITA. (UPM).

Power Concepts : Prezi