# Problem of Apollonius : ccc

Any of the problems of tangents that are included under the name "Apolonio problems" can be reduced to one of the variants studied the most basic of them all: the fundamental problem of tangents (PFT).

In this case we will study what we call "Apolonio Case ccc", namely, If the problem of tangents in which the data are given by conditions tangents three circumferences (ccc).

# 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.

# Polar of a point with respect to two lines

The concept of polarity is linked to the harmonic separation.

This concept is Basic for the determination of the fundamental elements of conics, as its Center, conjugate diameters, axes ….

It will allow to establish new transformations which include homographies and correlations of great importance.

# What is an involution in geometry?

In geometry, we speak often with terms that, in some cases, they are not sufficiently important in everyday language. This leads to create barriers in the interpretation of some simple concepts.

One of the terms that I have been asked several times in class is the of “Involution”. We define the involution.

What is an involution?

# Metric geometry: Loci. Arco able : Problema II Solución

Vamos a resolver un sencillo problema planteado anteriormente en el que deberemos determinar un lugar geométrico básico para la determinación de su solución, un problema en el que hay que encontrar un punto del plano que cumpla unas condiciones geométricas dadas.

La intersección de dos lugares geométricos planos nos determinará un número finito de puntos que serán las posibles soluciones del problema.

# Metric geometry: Loci. Arco able : Problema II

Las técnicas de solución de problemas basadas en la intersección de lugares geométricas se suelen asociar a problemas sencillos de la geometría clásica.

En estos casos es el planteamiento de la solución lo que entraña la mayor complejidad, ya que los lugares geométricos derivados suelen ser elementos geométricos sencillos.
Determinar un punto P desde el que se observe bajo el mismo ángulo a los tres lados de un triángulo ABC.

# 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.

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

Los problemas básicos de geometría métrica tienen una especial belleza. Son adecuados para introducir a los alumnos en el arte del análisis en esta disciplina.

Uno de los problemas propuestos en el examen de Selectividad de Septiembre de 2014 plantea la obtención de una figura geométrica simple, un cuadrado, cuyos vértices se encuentran sobre elementos geométricos dados.

# The problem with the pool table: Solución

By raising the issue of the pool table, that is to hit one of the two balls that are on the table (A for example) , so that it impacts the other (la B) previously given in one of the bands (edges) Table, flipping the closed problem to a simple bounce case.

We can generalize the problem considering that you can give, before impact with the second ball, a given number of impacts with the bands (lateral edges) Table.

# Equivalent figures : Square equivalent [I]

Geometric figures can be compared with each other by reference for this comparison both its shape and its size.

Based on the different combinations that can be found in these comparisons will classify in:

Similar forms: Have the same shape but different size
Equivalent forms: They have different but equal size (Volume of the area)
Congruent shapes: Have the same shape and size (equal)
Overall, to obtain a form equivalent to another given, use an equivalent square as intermediate between two equivalent figures. Thus, first discuss how to obtain a square equivalent to a geometric figure.