These problems are solved by intersecting loci.
En particular, if we consider the line as infinite radius circumference, We will therefore in the case studied straight with angular determination conditions.
Condition angle with respect to straight
We begin the analysis with tangency conditions (zero angle) to determine the locus of centers of circumferences of known radius that is tangent to a straight r. Subsequently generalize these loci for any angle of incidence.
To determine a circle need three geometric constraints. In the problem we proposed as data the radius of the circle and the tangency condition, with a level of freedom to define said circumference.
We therefore infinite solutions and, accordingly, a locus for their centers.
Suppose that we seek through a point T concrete in the line tangent r. Downtown O will be in the perpendicular to r through point T, the distance R (radius of the circle). If we move the point T along r find infinite solutions and centers, accordingly, the locus of the centers LG is a line parallel to the previous distance R.
Actually we have two possible loci, since the distance R we have taken from the point of tangency T can be in both directions perpendicular to the direction.
If instead of considering a tangent constraint we use an angular condition, the problem does not differ much.
Determine a solution (which passes through a point P) and generalize the locus. For this, point P look straight t forming with the straight r the angular condition. This straight t is the tangent to the circle at the point P and its center will be on her and the perpendicular distance R.
Again we find two straight as possible loci for possible solutions centers.
Condition angle with respect to circles
If the condition is angled with respect to a circumference, the procedure for determining the locus of the centers is similar. We will seek a solution through a point of the circle and determine the locus.
If the condition is tangent, at a point T determine any tangent t and find the center distance R in the direction perpendicular to said tangent. We see that in this case geometric them are places of concentric circles with which we have given as given, c, radii of the sum or difference of radii c and value R.
If the condition is any angle we determine the tangent to c at any point P and get a line passing through this point and form the given angle. This line will be tangent to the solution we seek and find its center in the perpendicular distance R.
In the above figure has only been given one of the two loci. Another result we get marking a line with the angular condition in the opposite direction.
Note that the condition of passing through a point is the same as considering that the data has a zero radius circle, analogously to think that a condition regarding a line is assumed that the radius is of infinite length.
Application to problem solving
We can solve different problems in which the radius of the desired circle is known by the intersection of the loci we have seen. We will need to impose two additional geometric conditions to complete the problem:
- Passing through two points
- Passing through a point and are tangent to a straight
- Passing through a point and are tangent to a circle
- Passing through a point and forming an angle with a straight
- Passing through a point and forming an angle with a circumference
- That are tangent to the straight
- What are the tangents of circles
- That are tangent to a line and a circle
- Forming an angle with a line and another straight line tangent to
- Forming an angle with a circle and are tangential to the other straight
- Forming an angle with a straight and one with another straight
- Forming an angle with a circumference and the other with another straight
- Forming an angle with a circumference and the other with another circle
Intersection of loci
Consider finally an application example of the statements that apply in the intersection of these loci in resolution.
Consider the following problem:
Determining known radius circumferences that are tangent to a straight line and a circumference
With the tangency condition and the given radius would obtain the corresponding loci.
Determine the intersection points of said loci to be the centers of the circumferences searched
We see that the number of solutions depends on the number of intersection points, Accordingly the relative positions of the data.