Metric geometry : Problema fundamental de tangencias : PPc [II]

Called fundamental problem of tangents puede presentarse con condiciones de tangencia respecto de una circunferencia, instead of a straight.

Conceptually we can assume that the above is a particular case of this, if we consider the straight as a circle of infinite radius.

In both cases therefore apply similar reasoning for resolution, based on the concepts learned in power.

Resolveremos el segundo caso de estudio enunciando el problema como:

Determine the circles through the points A and B y son tangentes a la circunferencia c

Enunciado del problema fundamental de tangencias

Analysis of the fundamental problem of tangents

In figure analysis shows that the circumference s can be one solution to the problem as it passes through points A and B y es tangente a la circunferencia c. In this figure ,in which the circumference represent solution we are looking for, we can determine properties that serve to derive a construction that allows us to determine it.

Se ha representado también otra circunferencia auxiliar (línea de trazos) que pasa por los puntos A and B y que corta a c en los puntos C and D.

Analysis of the fundamental problem of tangents

Straight A-B and C-D intersect at a point P que es el centro radical de las tres circunferencias y por lo tanto tiene igual potencia respecto de ellas, esto se puede expresar como:

Potencia del centro radical

From the above expression we deduce that if we get the segment value PT (Power root) we get the point T de tangencia entre c and s y el problema se reduce a determinar la circunferencia que pasa por tres puntos: A, B and T (its center will be at the intersection of two bisectors).

Resolving the problem.

Determine the value of the power by one of the constructions used to solve proportional means:

• Height theorem
• Theorem catheter
• Power

As the power point P with respect to any circle through the points A and B is the same, We can use an auxiliary circle of any radius passing through these points, as shown in the center figure O1, located on the bisector of A and B.

The power value obtaining determine the tangent segment from P this auxiliary circle; for this, build a arc able 90 grados the segment PO1

Resolución del problema fundamental de tangencias

The segment value tangency ( P-T1) lo llevaremos sobre la circunferencia c to determine the point Ta of tangency with a simple twist of center P.

Solución del problema fundamental de tangencias

Solution Number

Dependiendo de la dirección (lado de la circunferencia c) en que situemos el segmento PT get one or other of the two possible solutions.

Número de soluciones del problema fundamental de tangencias

Metric geometry