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Metric geometry : Radical axis of two circles

eje_radicalThe loci used to determine the solution of problems with geometric constraints.

Among the conditions used are the angular nature and among them the orthogonality.

Given the circles coplanarias, infinity simply set the circumferences that intersect orthogonally are grouped into a named set corradicales beam circumferences; These circles are centered on a line called radical axis.

The radical axis of two circles is the locus of points in the plane

  • which are centers of circles orthogonal tells circles
  • having equal power tells about the circumferences
  • from which can be drawn equal length segments tangent to the circumferences

To determine this locus, radical axis, we will rely on a shape analysis comprises two circumferences which are orthogonally intersected by the desired.


We see triangles that meets, applying Pythagoras, the following relationships:


from which we can obtain

diferencia de cuadrados

as we have seen in studying the locus of the difference of squares of distances from two fixed points, is a straight. This line is called radical axis of two circles.

Radical center of three circles

We see that by imposing two restrictions of orthogonality is determined a locus for the centers of the solutions that meet. If we introduce a third condition we obtain a unique solution we can get through the aforementioned intersection loci.

The CR radical center three coplanar circles is a point in its plane:
  • is intersection of the three axes of the circumferences radicals
  • has equal power with respect to these circles
  • is center circle orthogonal to these circles
  • from which can be drawn tangent segments of equal length to the three circles
The radical center can be obtained by the intersection of two radical axes
centro radical de tres circunferencias

Radical center of three circles

Metric Geometry