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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.

La condición depaso por un punto Pnos permitirá en todos los casos realizar una inversión de centro este punto para simplificar la naturaleza de lasolución del problema ya que si invertimos el sistema con centro ese punto nuestra solución buscada se convertirá en la inversa de una circunferencia que pasa por el centro de inversión: a line.

En los casos en los que se den condiciones de isogonalidad (igual ángulo con dos circunferencias) la solución será doble en la inversión que relacione a las dos circunferencias dato y, accordingly, orthogonal to the self-inverting (caso positivo). Este modelo de análisis conducirá a la búsqueda de circunferencias ortogonales de nuevo.

Si el punto de paso se encuentra en alguna de las circunferencias podremos cambiar las condiciones de angularidad respecto de la circunferencia por una tangencia respecto de una recta que pase por el punto y forme el ángulo solicitado con la circunferencia. En este caso también podremos buscar circunferencias pertenecientes a un haz parabólico.

Los siguientes problemas son de fácil resolución con las estrategias mencionadas. Podemos enunciar todos los problemas con un mismo enunciado general:

Determinar la circunferencia que forma un ángulo a con c1 , b con c2 y pasa por el punto P

Case 1

Dos condiciones de tangencia y un punto de paso situado sobre una de las circunferencias.

Case 2

Dos condiciones angulares iguales (isogonalidad) y un punto de paso situado sobre una de las circunferencias.

Case 3

Dos condiciones angulares iguales (isogonalidad), in particular orthogonality (membership conjugate beam) y un punto de paso situado sobre una de las circunferencias.

Case 4

Two different angular conditions, particularly tangency and 30 degrees and a crossing point located on one of the circumferences (Parabolic beam or line tangent to).

Case 5

Two tangency conditions (Isogonalidad) y un punto de paso situado sobre una de las circunferencias.

Case 6

Two tangency conditions (Isogonalidad) and a point of free passage (The system can be reversed from the center of similarity between the circumferences or from P)

 

 

Below you can see a PDF file with the proposal of previous exercises.

Download (PDF, 52KB)

We will continue with new "mental gymnastics tables for geometry" in future entries.

Metric geometry

 

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