Radical Axis - Geometric Construction

Geometric Construction

The radical axis of two circles A and B can be constructed by drawing a line through any two of its points. Such a point can be found by drawing a circle C that intersects both circles A and B in two points. The two lines passing through each pair of intersection points are the radical axes of A and C and of B and C. These two lines intersect in a point J that is the radical center of all three circles, as described above; therefore, this point also lies on the radical axis of A and B. Repeating this process with another such circle D provides a second point K. The radical axis is the line passing through both J and K.

A special case of this approach is carried out with antihomologous points from an internal or external center of similarity. Consider two rays emanating from an external homothetic center E. Let the antihomologous pairs of intersection points of these rays with the two given circles be denoted as P and Q, and S and T, respectively. These four points lie on a common circle that intersects the two given circles in two points each. Hence, the two lines joining P and S, and Q and T intersect at the radical center of the three circles, which lies on the radical axis of the given circles. Similarly, the line joining two antihomologous points on separate circles and their tangents form an isosceles triangle, with both tangents being of equal length. Therefore, such tangents meet on the radical axis.